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Research article Special Issues

Discrete stage-structured tick population dynamical system with diapause and control


  • Received: 04 July 2022 Revised: 02 August 2022 Accepted: 21 August 2022 Published: 05 September 2022
  • A discrete stage-structured tick population dynamical system with diapause is studied, and spraying acaricides as the control strategy is considered in detail. We stratify vector populations in terms of their maturity status as immature and mature subgroups. The immature subgroup is divided into two categories: normal immature and diapause immature. We compute the net reproduction number R0 and perform a qualitative analysis. When R0<1, the global asymptotic stability of tick-free fixed point is well proved by the inherent projection matrix; there exists a unique coexistence fixed point and the conditions for its asymptotic stability are obtained if and only if R0>1; the model has transcritical bifurcation if R0=1. Moreover, we calculate the net reproduction numbers of the model with constant spraying acaricides and periodic spraying acaricides, respectively, and compare the effects of the two methods on controlling tick populations.

    Citation: Ning Yu, Xue Zhang. Discrete stage-structured tick population dynamical system with diapause and control[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 12981-13006. doi: 10.3934/mbe.2022606

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  • A discrete stage-structured tick population dynamical system with diapause is studied, and spraying acaricides as the control strategy is considered in detail. We stratify vector populations in terms of their maturity status as immature and mature subgroups. The immature subgroup is divided into two categories: normal immature and diapause immature. We compute the net reproduction number R0 and perform a qualitative analysis. When R0<1, the global asymptotic stability of tick-free fixed point is well proved by the inherent projection matrix; there exists a unique coexistence fixed point and the conditions for its asymptotic stability are obtained if and only if R0>1; the model has transcritical bifurcation if R0=1. Moreover, we calculate the net reproduction numbers of the model with constant spraying acaricides and periodic spraying acaricides, respectively, and compare the effects of the two methods on controlling tick populations.



    Numerical computations on the real (compact) Stiefel manifold viewed as the embedded submanifold Stn,k={XRn×kXX=Ik} of Rn×k arise in many branches of applied mathematics like numerical linear algebra and, moreover, in the engineering context, as well. Beside interpolation problems [1], we mention the following examples which are closely linked to optimization. For instance, the symmetric eigenvalue problem can be formulated as an optimization problem on the Stiefel manifold [2]. Moreover, one encounters optimization problems on Stn,k in connection with machine learning [3], multivariate data analysis [4] and computer vision [5,6]. These problems can be tackled by Riemannian optimization methods, see e.g. [2,7,8,9]. An essential part of their design is the choice of an appropriated Riemannian metric [7, Chap. 1]. The Euclidean metric, see e.g. [2], and the so-called canonical metric, see e.g. [10], are well-known, common choices for the Stiefel manifold. For these two metrics, explicit formulas for Riemannian gradients and Riemannian Hessians of smooth functions are known. Such formulas are desirable for the application of several Riemannian optimization methods.

    However, there is no reason to restrict to one of these two metrics. In principle, the performance of a Riemannian optimization method could be improved by choosing an alternative metric adapted to the particular function under consideration. For example, the dependence of the speed of convergence of a Riemannian optimization method on the Riemannian metric is investigated in [11] on "Riemannian preconditioning". Moreover, a family of metrics on the generalized Stiefel manifold is introduced in [11] which differs from the family of metrics on Stn,k discussed here.

    In this paper, we investigate a 2k-parameter family of pseudo-Riemannian metrics on Stn,k from an extrinsic point of view. This family does not coincide with the family of metrics considered in [12]. Nevertheless, it contains the Euclidean metric and the so-called canonical metric. In addition, the whole one-parameter family which has been recently introduced in [13] is included. An emphasize is put on deriving explicit formulas for gradients and Hessians suitable for applying them in connection with Riemannian optimization methods. In particular, specific results of the conference paper [14] are reproduced as special cases.

    Next we give an overview of this text which is kept as self-contained as possible. We start with endowing Rn×k with a family of covariant 2-tensors depending on 2k parameters, which are invariant under the O(n)-left action on Rn×k by matrix multiplication from the left. For suitable choices of these parameters, the corresponding 2-tensor induces a pseudo-Riemannian metric on an open subset U of Rn×k such that Stn,kU becomes a pseudo-Riemannian submanifold of U. Hence it makes sense to consider the normal bundle of Stn,k and the orthogonal projections onto the tangent spaces of Stn,k which can be described by explicit formulas.

    In order to put this extrinsic approach into context to existing works on families of metrics on the Stiefel manifold we also consider Stn,k, equipped with our family, as a pseudo-Riemannian reductive homogeneous SO(n)-space. This point of view shows that, for the Riemannian case, the family of metrics which is discussed in this text, is partially contained in the family considered in the work [15] on Einstein metrics. Nevertheless, at least to our best knowledge, the family of metrics on Stn,k considered in this paper has never been treated before from an extrinsic point of view.

    After this short detour, we come back to the extrinsic approach. We derive an explicit expression for the spray S:TStn,kT(TStn,k) associated with the metric. To this end, we exploit a well-known fact, see e.g. [16, Sec. 7.5] for the Riemannian case. The metric spray of a pseudo-Riemannian manifold coincides with the Lagrangian vector field on its tangent bundle associated with the kinetic energy defined by means of the pseudo-Riemannian metric. This allows for computing the metric spray on the tangent bundle TU, where URn×k is the open set of which Stn,k is a pseudo-Riemannian submanifold. Eventually, by using a result from [16, Sec. 8.4] on constrained Lagrangian systems, combined with the explicit expression for the orthogonal projections, the metric spray on TStn,k is computed. As a by-product, the geodesic equation is obtained as an explicit second order matrix valued ordinary differential equation (ODE).

    Next we derive expressions for pseudo-Riemannian gradients and pseudo-Riemannian Hessians of smooth functions on Stn,k involving only "ordinary" matrix operations. Using the formula for the orthogonal projection onto tangent spaces, we derive an explicit formula for pseudo-Riemannian gradients. Moreover, since we have an expression for the geodesic equation as explicit second order matrix valued ODE, we obtain an explicit formula for pseudo-Riemannian Hessians, too. The expression for the pseudo-Riemannian gradient is valid for all metrics in the 2k-parameter family, while, for the pseudo-Riemannian Hessian, we restrict ourself to a subfamily depending on (k+1)-parameters in order to obtain formulas which are not too complicated. This (k+1)-parameter subfamily still contains the Euclidean metric and the canonical metric as well as the one-parameter family from [13].

    Finally, a formula for the second fundamental form of Stn,k considered as pseudo-Riemannian submanifold of an open URn×k is derived. We give a concrete expression for the second fundamental form with respect to the metrics in the (k+1)-parameter subfamily. By means of the Gauß formula, an explicit matrix-type formula for the Levi-Civita covariant derivative is obtained.

    Throughout this text, except for Section 3.4, we view the real (compact) Stiefel manifold Stn,k as an embedded submanifold of the real (n×k)-matrices Rn×k which is given by

    Stn,k={XRn×kXX=Ik}Rn×k,1kn. (2.1)

    We point out that Stn,k is a proper subset of Rn×k although the inclusion is denoted by Stn,kRn×k. In the sequel, we often denote proper inclusions by "". The symbol "" is only used if we want to emphasize that an inclusion is not an equality. The tangent bundle of Stn,k is denoted by TStn,k which is considered as a submanifold of TRn×kRn×k×Rn×k. More generally, for a manifold M, we denote by TM and TM its tangent and cotangent bundle, respectively. In the sequel, if not indicated other-wise, we identify Rn×k with its dual space (Rn×k) via the linear isomorphism

    Rn×k(Rn×k),Vtr(V())=(Wtr(VW)) (2.2)

    induced by the Frobenius scalar product. The following characterization of the tangent space of Stn,k at XStn,k considered as subspace of Rn×k is used frequently

    TXStn,k={VRn×kXV=VX}Rn×k. (2.3)

    We write

    O(n)=Stn,n={RRn×nRR=RR=In} (2.4)

    for the orthogonal group and

    SO(n)={RRn×kRR=RR=In and det(R)=1} (2.5)

    for the special orthogonal group. Their Lie algebras coincide and are denoted by

    so(n)={ξRn×nξ=ξ}. (2.6)

    Moreover, we write

    skew:Rn×nso(n)Rn×n, A12(AA) (2.7)

    for the projection onto so(n) whose kernel is given by the set of symmetric matrices Rn×nsym. The O(n)-left action on Rn×k by matrix multiplication from the left is denoted by

    Ψ:O(n)×Rn×kRn×k,(R,X)RX. (2.8)

    By restricting the second argument of Ψ one obtains the O(n)-action

    O(n)×Stn,kStn,k,(R,X)RX (2.9)

    on Stn,k from the left which we denote by Ψ, as well. It is well-known that this O(n)-action on Stn,k is transitive. For fixed RO(n) we denote the diffeomorphisms induced by the actions from (2.8) and (2.9)

    Rn×kXRXRn×k and Stn,kXRXStn,k (2.10)

    both by ΨR.

    If URn×k is some subset, we write

    ιU:URn×k (2.11)

    for the canonical inclusion of U into Rn×k. Moreover, the canonical inclusion of Stn,k into Rn×k is often denoted by

    ι:Stn,kRn×k (2.12)

    for short.

    Next let pr:FM be a vector bundle over a manifold M with dual bundle F. The smooth sections of F are denoted by Γ(F). Moreover, we denote by F, S(F) and Λ(F) the -th tensor power, the -th symmetrized tensor power and the -th antisymmetrized tensor power of F, respectively. In addition, we write End(F)FF for the endormorphism bundle of F. The vertical bundle of F is denoted by Ver(F)TF.

    Let f:MN be a smooth map between manifolds and let αΓ((TN)) be a covariant tensor field on N. The pullback of α by f is denoted by fα. If α is a differential form, i.e. αΓ(Λ(TM)), the exterior derivative of α is denoted by d α. The tangent map of f is denoted by Tf:TMTN. If f is a map between (open subsets of) finite dimensional R-vector spaces, we write Df(X)V for the derivative of f at X evaluated at V. Sometimes, the tangent map of a smooth map f between arbitrary manifolds at the point X evaluated at a tangent vector V is denoted by Df(X)V, as well.

    Next let MRn×k be a submanifold. A vector field V:MTMRn×k×Rn×k is often implicitly identified with the map MRn×k defined by its second component which we denote by V, as well, i.e. the "foot point" XM is suppressed in our notation. If SΓ(T(TM)) is a vector field on TM, we view it as a map S:TMT(TM)(Rn×k)4 usually not suppressing the "foot point" (X,V)TM.

    For a smooth function F:Rn×kR we write F(X) for the gradient of F at XRn×k with respect to the Frobenius scalar product, i.e. the unique matrix F(X)Rn×k with

    d F|X(V)=tr((F(X))V) (2.13)

    for all VRn×k. Furthermore EijRn×k denotes the matrix whose entries fulfill (Eij)f=δifδj for all f{1,,n} and {1,,k} with δif and δj being Kronecker deltas.

    Finally, following the convention in [17, Chap. 2], a scalar product is a non-degenerated symmetric bilinear form. Moreover, an inner product is a positive definite symmetric bilinear form.

    We start with investigating a 2k-parameter family of symmetric covariant 2- tensors on Rn×k. For certain choices of these parameters, it defines a pseudo-Riemannian metric on an open subset URn×k such that Stn,kU becomes a pseudo-Riemannian submanifold of U.

    We introduce a 2k-parameter family of symmetric covariant 2-tensors on Rn×k.

    Lemma 3.1. Let D=diag(D11,,Dkk)Rk×k and E=diag(E11,,Ekk)Rk×k be both diagonal. Then the point-wise definition

    V,WD,EX=tr(VWD)+tr(VXXWE) (3.1)

    with XRn×k and V,WTXRn×kRn×k yields a smooth covariant 2-tensor ,D,E()Γ(S2(TRn×k)) which is invariant under the O(n)-action Ψ defined in (2.8).

    Proof. Obviously, (3.1) defines a smooth covariant 2-tensor. Let RO(n). Then ΨR,D,E()=,D,E() holds due to

    DΨR(X)V,DΨR(X)WD,EΨR(X)=RV,RWD,ERX=V,WD,EX

    for XRn×k and V,WTXRn×kRn×k showing the Ψ-invariance of ,D,E().

    Remark 3.2. Observe that the diagonal entry EiiR of the diagonal matrix E=diag(E11,,Ekk)Rk×k shall not be confused with the matrix EiiRn×k introduced at the end of Section 2. In the sequel, it should be clear by the context how the symbol Eii has to be understood.

    Remark 3.3. Let E=0. Then ,D,E()Γ(S2(TRn×k)) becomes independent of XRn×k. Hence we may identify ,D,0() with the symmetric bilinear form

    ,D:Rn×k×Rn×kR,(V,W)V,WD=tr(VWD). (3.2)

    If we want to emphasize that ,D is a symmetric bilinear form on Rn×k, we denote it by ,DRn×k.

    Remark 3.4. The pull-back ι,D,E()Γ(S2(TStn,k)) of ,D,E() with ι:Stn,kRn×k simplifies for the following values of k:

    1. For k=n one has Stn,n=O(n). Thus for XO(n) and V,WTXRn×kRn×k one obtains

    V,WD,EX=tr(VW(D+E))=V,WD+E (3.3)

    due to XX=XX=In, i.e. ι,D,E()=,D+E holds.

    2. For k=1 one has Stn,1=Sn1Rn. Using XV=0 for all XSn1 and VTXSn1 yields

    V,WD,EX=V,WD, (3.4)

    i.e. ι,D,E()=,D holds.

    Remark 3.5. The pull-back ι,D,E()Γ(S2(TStn,k)) yields well-known metrics on Stn,k for certain choices of D and E:

    1. For D=Ik and E=0 one obtains the Euclidean metric, see e.g. [10], [18, Sec. 23.5] or [2]

    2. Setting D=Ik and E=12Ik yields the canonical metric, see e.g. [10] or [18, Sec. 23.5]

    3. For D=2Ik and E=νIk with ν=2α+1α+1 and αR{1} the metric ,D,E() reproduces a one-parameter family which has been introduced in [13], see in particular [13, Eq. (55)].

    In order to investigate ,D,E()Γ(S2(TRn×k)) and its pull-back to Stn,k we first list some properties of ,D.

    Lemma 3.6. Let D=diag(D11,,Dkk)Rk×k be diagonal. The following assertions are fulfilled:

    1. The symmetric bilinear form ,D:Rn×k×Rn×kR is a scalar product iff D is invertible.

    2. The bilinear form ,D:Rn×k×Rn×kR is an inner product iff Dii>0 holds for all i{1,,k}.

    3. Assume that D is invertible. Then ,D:Rk×k×Rk×kR induces a scalar product on so(k) iff

    Dii+Djj0 (3.5)

    holds for all i,j{1,,k}. This condition is always satisfied for k=1.

    4. Let k2. Then ,D|so(k)×so(k):so(k)×so(k)R defines an inner product on so(n) iff

    Dii+Djj>0 (3.6)

    holds for all 1i<jk. For k=1, this bilinear form defines always an inner product.

    Proof. Let EijRn×k denote the matrix whose entries fulfill (Eij)f=δifδj. Clearly, the set

    B={Eiji{1,,n}andj{1,,k}}

    defines a basis of Rn×k. Thus it suffices to show that for all EijB the associated linear forms

    Rn×kR,VEij,VD (3.7)

    are non-zero iff D is invertible. We have

    Eij,VD=tr(EijVD)=VijDjj (3.8)

    with V=(Vij)Rn×k. Equation (3.8) implies that D is invertible iff the linear forms in (3.7) are non-vanishing for all i{1,,n} and j{1,,k} showing Claim 1.

    Next we prove Claim 2. Let 0V=(Vij)Rn×k. Then V,VD>0 holds iff Dii>0 for i{1,,k} due to

    V,VD=tr(VVD)=ki=1nj=1V2jiDii.

    We now prove Claim 3. For k=1 the assertion is trivial due to dim(so(1))=0. For k2 the set {EijEji1i<jk} is a basis of so(k). Thus ,D induces a scalar product on so(k) iff the linear forms

    so(k)R,AEijEji,AD

    are non-vanishing for all 1i<jk. Writing A=(Aij)=(Aji)so(k) we compute

    EijEji,AD=Eij,ADEji,AD=AijDjjAjiDii=Aij(Djj+Dii)

    showing that ,D defines a scalar product on so(k) iff

    Dii+Djj0,i,j{1,,k}

    holds. Here we exploited that Dii+Dii0 is automatically fulfilled because D is invertible.

    It remains to prove Claim 4. The case k=1 is trivial due to so(1)={0}. Thus assume k2. Let A=(Aij)so(k). Exploiting Aij=Aji we calculate

    A,AD=12tr(AAD)+12tr(AAD)=12ki,j=1A2ij(Dii+Djj). (3.9)

    Using Aii=0 we conclude that A,AD>0 holds for all 0Aso(k) iff Dii+Djj>0 is fulfilled for all 1i<jk.

    The next lemma shows that ,D,E() induces a pseudo-Riemannian metric on the Stiefel manifold for certain choices of D and E.

    Lemma 3.7. Let D=diag(D11.,Dkk)Rk×k and E=diag(E11,,Ekk)Rk×k be both diagonal and let XStn,k. Then the following assertions are fulfilled:

    1. Let 1k<n. The bilinear form

    ,D,EX:TXRn×k×TXRn×kRn×k×Rn×kR (3.10)

    is a scalar product iff D and D+E are both invertible. For k=n the bilinear form in (3.10) defines a scalar product iff D+E is invertible.

    2. Assume that (3.10) defines a scalar product. Then the pull-back ι,D,E() to Stn,k defines a pseudo-Riemannian metric on Stn,k, i.e.

    ,D,EX:TXStn,k×TXStn,kR (3.11)

    is a scalar product on TXStn,k, iff the condition

    Dii+Eii+Djj+Ejj0,i,j{1,,k} (3.12)

    holds.

    3. Assume that (3.10) defines a scalar product. For 2kn1 the symmetric covariant 2-tensor ι,D,E()Γ(S2(TStn,k)) is a Riemannian metric on Stn,k, i.e.

    ,D,EX:TXStn,k×TXStn,kR (3.13)

    is an inner product on TXStn,k, iff the conditions Dii>0 for all i{1,,k} and

    Dii+Eii+Djj+Ejj>0,1i<jk (3.14)

    are fulfilled. For k=1 one obtains a Riemannian metric iff D11>0 holds. For k=n the tensor ι,D,E() defines a Riemannian metric iff Dii+Eii+Djj+Ejj>0 holds for all 1i<jn.

    Proof. Since the O(n)-left action Ψ on Rn×k defined in (2.8) is isometric with respect to ,D,E() by Lemma 3.1 and, moreover, Ψ restricts to a transitive action on Stn,k it suffices to prove the claims for a single point X0Stn,k.

    We first consider the case k=n. Then ,D,EX=,D+E holds for all XStn,n=O(n) by Remark 3.4, Claim 1. Hence ,D,EX is non-degenerated iff D+E is invertible according to Lemma 3.6, Claim 1. Next we consider the case 1k<n. We choose X0=In,k, where

    In,k=[Ik0]Stn,k,

    and write

    V=[V1V2]Rn×kandW=[W1W2]Rn×k

    with V1,W1Rk×k and V2,W2R(nk)×k. By this notation and identifying TXRn×kRn×k we calculate

    V,WD,EIn,k=tr([V1V2][W1W2]D)+tr([V1V2][Ik000][W1W2]E)=tr(V1W1(D+E))+tr(V2W2D). (3.15)

    By (3.15) and Lemma 3.6, Claim 1, the bilinear form ,D,EIn,k defines a scalar product on TXRn×k iff D and D+E are both invertible.

    Next we assume that D and D+E are choosen such that ,D,EX defines a scalar product on TXRn×k for each XStn,k. We now prove Claim 2 for 1kn1. To this end, it is sufficient to show that

    ,D,EIn,k:TIn,kStn,k×TIn,kStn,kR (3.16)

    is a scalar product iff (3.12) holds. The tangent space TIn,kStn,k is given by

    TIn,kStn,k={[V1V2]|V1so(k) and V2R(nk)×k}TIn,kRn×kRn×k, (3.17)

    see e.g. [10, Sec. 2.2.1]. Thus we may write V,WTXStn,k as

    V=[V1V2]Rn×k andW=[W1W2]Rn×k

    with V1,W1so(k) and V2,W2R(nk)×k. We now obtain

    ιV,WD,EIn,k=tr(V1W1(D+E))+tr(V2W2D) (3.18)

    analogously to (3.15). Clearly, Equation (3.18) defines a scalar product on TIn,kStn,k iff

    so(k)×so(k)R,(V1,W1)tr(V1W1(D+E))

    yields a scalar product on so(k) and

    R(nk)×k×R(nk)×kR,(V2,W2)tr(V2W2D)

    defines a scalar product on R(nk)×k. By applying Lemma 3.6, Claim 3 we obtain the desired result. Next we consider the case k=n. By exploiting the O(n)-invariance of ,D,E() and TInStn,n=so(n) as well as ,D,E()=,D+E for k=n, Claim 2 follows by Lemma 3.6, Claim 3.

    It remains to prove Claim 3. We first consider the case 2kn1. Since the bilinear form on TIn,kStn,k induced by ,D,E() is given by (3.18), the desired result is a consequence of Lemma 3.6, Claim 2 and Lemma 3.6, Claim 4. For k=1, we observe that ι,D,E() is independent of E due to XV=0 for all XStn,1 and VTXStn,1, see also Remark 3.4, Claim 2. Hence (3.18) implies that ,D,E() is positive definite iff

    R(nk)×k×R(nk)×kR,(V2,W2)tr(V2W2D)

    is positive definite. The desired result follows by Lemma 3.6, Claim 2. For k=n, the assertion holds due to ,D,EX=,D+E for all XStn,n=O(n) by Lemma 3.6, Claim 3.

    The next lemma generalizing [14, Lem. 2] shows that there is an open neigbourhood URn×k of Stn,k such that Stn,k(U,ιU,D,E()) is a pseudo-Riemannian submanifold. This fact is crucial for the following discussion.

    Lemma 3.8. Let D,ERk×k be both diagonal such that for each XStn,k

    ,D,EX:TXRn×k×TXRn×kR (3.19)

    defines a scalar product on TXRn×kRn×k which induces a scalar product on TXStn,kTXRn×k. Then there exists an open neighbourhood URn×k of Stn,k such that ιU,D,E()Γ(S2(TU)) is a pseudo-Riemannian metric on U and (Stn,k,ι,D,E()) is a pseudo-Riemannian submanifold of (U,ιU,D,E()).

    Proof. We identify ,D,E()Γ(S2(TRn×k)) with the continuous map

    φ:Rn×kS2((Rn×k)),X,D,EX=((V,W)V,WD,EX).

    The bilinear form φ(X)=,D,EXS2((Rn×k)) is a scalar product for all XStn,k by assumption. Hence, by the continuity of φ, there is an on open neighbourhood UX of X in Rn×k such that φ(˜X)S2((Rn×k)) is non-degnerated for all ˜XUX. We set

    U=XStn,kUX.

    Then URn×k is open as a union of open sets and fulfills Stn,kU by definition. Moreover, φ(˜X) is non-dengenerated for all ˜XU by construction. Hence ιU,D,E() defines a pseudo-Riemannian metric on U such that Stn,k(U,ιU,D,E()) is a pseudo-Riemannian submanifold.

    Obviously, the inclusion Stn,kU from Lemma 3.8 is always proper since Stn,k is closed in Rn×k while U is open in Rn×k.

    Notation 3.9. From now on, unless indicated otherwise, pull-backs of ,D,E() to submanifolds of Rn×k are suppressed in the notation.

    In the case k=n, the 2k-parameter family of covariant 2-tensors ,D,E() is actually a k-parameter family by Remark 3.4, Claim 1. Indeed, ,D,E() depends only on D+E. Hence one may ask if there exits always such an over-parameterization.

    Lemma 3.10. Let D=diag(D11,,Dkk)Rk×k be some diagonal matrix. Then the following assertions are fulfilled:

    1. The bilinear form ,D:Rn×k×Rn×kR vanishes identically iff D=0 holds.

    2. The restriction ,D|so(k)×so(k):so(k)×so(k)R of ,D:Rk×k×Rk×kR fulfills the following assertions:

    (a) For k=1 one has ,D|so(k)×so(k)=0 for all DR1×1R.

    (b) For k=2 one has ,D|so(k)×so(k)=0 iff D11+D22=0 holds.

    (c) For k3 one has ,D|so(k)×so(k)=0 iff D=0 holds.

    Proof. Let EijRn×k the matrix whose entries fulfill (Eij)f=δifδj. Then

    Eij,VD=VijDjj,i{1,,n}, j{1,,k}, (3.20)

    where V=(Vij)Rn×k. Since ,D=0 holds iff the linear forms Eij,D:Rn×kR vanishes for all 1in and 1jk, the first claim follows by (3.20).

    Next, we consider ,D|so(k)×so(k):so(k)×so(k)R. Clearly, it vanishes for k=1 for all DR1×1 due to so(1)={0}.

    We now assume k2. Then ,D|so(k)×so(k):so(k)×so(k)R vanishes iff the linear forms

    EijEji,D:so(k)R (3.21)

    vanish for all 1i<jk. Writing A=(Aij)=(Aji)so(k) we obtain

    EijEji,AD=AijDiiAjiDjj=Aij(Dii+Djj).

    Thus the linear forms (3.21) are zero iff Dii+Djj=0 holds for all 1i<jk. For k=2 this is equivalent to D11+D22=0. It remains to consider the case k3. The conditions Dii+Djj=0 for all 1i<jk include the conditions

    D11+Dii=0D11=Dii for all2ik (3.22)

    and

    D(k1)(k1)+Dkk=0. (3.23)

    In particular D11=Dk1 and D11=Dkk holds. Plugging these identities into (3.23) yields

    D11D11=2D11=0D11=0.

    Hence (3.22) implies Dii=0 for all 2ik. Therefore ,D|so(k)×so(k)=0 iff D=0 as desired.

    The next lemma justifies calling ,D,E() a 2k-parameter family provided that 3kn1 holds.

    Lemma 3.11. Let

    Rk×kdiag={diag(D11,,Dkk)D11,,DkkR}Rk×k

    denote the k-dimensional real vector space of (k×k)-diagonal matrices. Moreover, define

    ψ:Rk×kdiag×Rk×kdiagΓ(S2(TStn,k)),(D,E),D,E(). (3.24)

    Then ψ is a linear map which fulfills the following assertions depending on k and n:

    1. For k=1=n, one has dim(im(ψ))=0 and ker(ψ)=R×R.

    2. For k=1 and n>1 one has dim(im(ψ))=1 and ker(ψ)={(0,E)ER}R×R.

    3. For k=2=n one has dim(im(ψ))=1 and

    ker(ψ)={((D11,D22),(E11,D11D22E11))D11,D22,E11R}R2×2diag×R2×2diag.

    4. For 2<k<n one has dim(im(ψ))=2k and ker(ψ)={0}Rk×kdiag×Rk×kdiag.

    5. For k=n>2 one has dim(im(ψ))=k and ker(ψ)={(D,D)DRk×kdiag}Rk×kdiag×Rk×kdiag.

    Proof. Clearly, the map ψ is linear. Next we define the linear map

    ˜ψ:Rk×kdiag×Rk×kdiagS2(TIn,kStn,k),(D,E),D,EIn,k.

    Obviously, for each (D,E)Rk×kdiag×Rk×kdiag one has (ψ(D,E))(In,k)=,D,EIn,k=˜ψ(D,E). Since ,D,E() is invariant under the transitive O(n)-action Ψ on Stn,k according to Lemma 3.1, this yields

    (D,E)ker(ψ)(D,E)ker(˜ψ). (3.25)

    Moreover, the equivalence

    (D,E)ker(˜ψ)(V,WD,EIn,k=0 for all V,WTIn,kStn,k) (3.26)

    is clearly fulfilled. We again write

    V=[V1V2]TIn,kStn,kandW=[W1W2]TIn,kStn,k

    with V1,W1so(k) and V2,W2R(nk)×k. By this notation and the description of ker(˜ψ) from (3.26), we study each case separately:

    1. Obviously, for k=1=n the claim ker(˜ψ)=R×R is correct due to TI1St1,1={0} implying dim(S2(TI1St1,1))=0.

    2. For k=1 and n>1 we have

    (˜ψ(D,E))(V,W)=tr(V1W1(D+E))+tr(V2W2D)=V1,W1D+E|so(1)×so(1)+V2,W2DR(n1)×1. (3.27)

    Clearly, Equation (3.27) vanishes iff D=0 holds independent of the value of D+E by Lemma 3.10. Hence the kernel of ψ is given by ker(˜ψ)={(0,E)ER}

    3. For k=2=n we have

    (˜ψ(D,E))(V,W)=V,WD+E|so(2)×so(2).

    Lemma 3.10 yields ˜ψ(D,E)=0 iff (D+E)11+(D+E)22=0 is fulfilled. Therefore we obtain

    ker(˜ψ)={((D11,D22),(E11,D11D22E11))D11,D22,E11R}.

    4. We now consider the case 3kn1. Then one has

    (˜ψ(D,E))(V,W)=tr(V1W1(D+E))+tr(V2W2D)=V1,W1D+E|so(k)×so(k)+V2,W2DR(nk)×k.

    By Lemma 3.10, we have ˜ψ(D,E)=0 iff D=0 and D+E=0 holds. Therefore the kernel of ψ is given by ker(˜ψ)={(D,E)Rk×kdiag×Rk×kdiagD=0=E}={0}.

    5. It remains to consider the case k=n3. We obtain

    (˜ψ(D,E))(V,W)=tr(VW(D+E))=V,WD+E|so(k)×so(k).

    for all V,WTInStn,n=so(n). Thus ˜ψ(D,E)=0 holds iff D+E=0 is fulfilled by Lemma 3.10. Hence the kernel of ˜ψ is given by ker(˜ψ)={(D,D)DRk×kdiag}.

    The equality ker(ψ)=ker(˜ψ) is satisfied according to (3.25). Moreover, we have

    dim(im(ψ))=dim(Rk×kdiag×Rk×kdiag)dim(ker(ψ))=2kdim(ker(˜ψ))

    as desired.

    Remark 3.12. Lemma 3.7, Claim 3 shows that the set of all parameters

    {(D,E)Rk×kdiag×Rk×kdiag,D,E() defines a pseudo-Riemannian metric on Stn,k}

    contains the non-empty subset {(D,E)Rk×kdiag×Rk×kdiagDii>0 and Eii>0 for all i{1,,k}} which is open in Rk×kdiag×Rk×kdiag. Moreover, the linear map ψ:Rk×kdiag×Rk×kdiagΓ(S2(TStn,k)) is injective for 2<k<n according to Lemma 3.11. This point of view justifies calling ,D,E() a 2k-parameter family at least for 2<k<n. For other choices of k and n one has rather a (dim(im(ψ)))-parameter family of metrics. However, ignoring this over parameterization, we call them 2k-parameter family, nevertheless.

    The Stiefel manifold Stn,k endowed with ,D,E()Γ(S2(TStn,k)) can be viewed as a pseudo-Riemannian submanifold of (U,,D,E()) with some suitable open URn×k by Lemma 3.8. Consequently, for any given point XStn,k, we may consider the orthogonal projection

    PX:TXRn×kTXStn,kRn×k,

    where TXRn×kRn×k is endowed with the scalar product ,D,EX. Moreover, it makes sense to consider the normal space NXStn,k=(TXStn,k)Rn×k with respect to ,D,EX:TXRn×k×TXRn×kR.

    Notation 3.13. From now on, unless indicated otherwise, we always assume that D,ERk×k are both diagonal matrices such that ,D,EX defines a scalar product on Rn×k for each XStn,k and ,D,E() induces a pseudo-Riemannian metric on Stn,k. In particular, we may assume that D and D+E are both invertible. In view of Lemma 3.7, Claim 1 this assumption is of no restriction. For the case k=n, we replace D by D+E and E by 0, if necessary.

    Lemma 3.14. Let D=diag(D11,,Dkk)Rk×k be invertible such that Dii+Djj0 holds for all i,j{1,,k}. Then the following assertions are fulfilled:

    1. The orthogonal complement of so(k) in Rk×k with respect to the scalar product ,D is given by

    so(k)D={ARk×kAD=(AD)}={ΛD1ΛRk×ksym}Rk×k. (3.28)

    Moreover, so(k)so(k)D=Rk×k holds.

    2. The orthogonal projection

    πD:Rk×kso(k)Rk×k,AπD(A) (3.29)

    onto so(k) with respect ,D is entry-wise given by

    πD(A)ij=1Dii+Djj(ADDA)ij=1Dii+Djj(AijDjjAjiDii),i,j{1,,k}. (3.30)

    Proof. We first determine so(k)D. To this end, we calculate

    so(k)D={ARk×kA,BD=0 for all Bso(n)}={ARk×ktr((AD)B)=0 for all Bso(n)}={ARk×kAD=(AD)Rk×ksym is symmetric }.

    Let ΛRk×ksym. Then (ΛD1)D=Λ=Λ=D(ΛD1) showing {ΛD1ΛRk×ksym}so(k)D. The equality so(k)D={ΛD1ΛRk×ksym} follows by counting dimensions. By Lemma 3.6, Claim 3 the assumptions on D ensure that ,D induces a scalar product on so(k). Hence so(k)so(k)D=Rk×k holds, see e.g. [17, Chap. 2, Lem. 23].

    It remains to prove Claim 2. To this end, we show im(πD)=so(k) and ker(πD)=so(k)D as well as πD|so(k)=idso(k). We first prove im(πD)so(n). Let A=(Aij)Rk×k. We compute

    ((πD(A)))ij=πD(A)ji=1Djj+Dii(AjiDiiAijDjj)=1Dii+Djj(AijDjjAjiDii)=πD(A)ij.

    for i,j{1,,k} showing im(πD)so(k). Moreover, for Aso(k), i.e. Aij=Aji, we have

    πD(A)ij=1Dii+Djj(AijDjj(Aij)Dii)=1Dii+DjjAij(Djj+Dii)=Aij.

    This yields πD(A)=A for all Aso(k), i.e. πD|so(k)=idso(k). Moreover, the inclusion im(πD)so(k) is in fact an equality. Next let Aso(k)D. Then AD=DA holds according to Claim 1 implying

    πD(A)ij=1Dii+Djj(ADDA)ij=0.

    Thus πD|soD=0 follows.

    The formula for πD can be rewritten in terms of the so-called Hadamard or Schur product. For matrices A,BRk×k, it is entry-wise defined by

    (AB)ij=AijBij,i,j{1,,k}. (3.31)

    Remark 3.15. Let μRk×k be defined entry-wise by

    μij=1Dii+Djj,i,j{1,,k}. (3.32)

    Then the projection πD:Rn×kso(k) from Lemma 3.14 can be rewritten as

    πD(A)=μ(ADDA),ARk×k. (3.33)

    Corollary 3.16. Let 0βR and define D=βIk. Then, for each ARk×k the map πD from Lemma 3.14 simplifies to

    πβIk(A)=12(AA)=skew(A). (3.34)

    Proof. The desired result follows by a straightforward calculation exploiting Dii=β0 for all i{1,,k}.

    We determine the normal spaces of Stn,k with respect ,D,E() generalizing [19, Chap. 1, Lem. 3.15] and [14, Lem. 3].

    Lemma 3.17. The normal space NXStn,k=(TXStn,k)TXRn×kRn×k at XStn,k with respect to ,D,EX is given by

    NXStn,k={XΛ(D+E)1Rn×kΛ=ΛRk×ksym}. (3.35)

    Proof. Clearly, the set {XΛ(D+E)1Rn×kΛ=ΛRk×ksym} is a linear subspace of Rn×k of dimension (k2+k)/2 being the image of the injective linear map

    Rk×ksymRn×k,ΛXΛ(D+E)1.

    Moreover, every matrix V=XΛ(D+E)1 with ΛRk×ksym is orthogonal to the tangent space TXStn,k. Indeed, we have for WTXStn,k

    V,WD,EX=tr((XΛ(D+E)1)WD)+tr((XΛ(D+E)1)XXWE)=tr(Λ(XW))=0

    due to Λ=Λ and XW=WX. Therefore {XΛ(D+E)1Rn×kΛ=ΛRk×ksym}NXStn,k follows. By counting dimensions, this inclusion is in fact an equality.

    Theorem 3.18. Let XStn,k. The orthogonal projection of TXRn×kRn×k onto TXStn,kRn×k with respect to ,D,EX is given by

    PX:Rn×kTXStn,kRn×k,VPX(V)=VXXV+XπD+E(XV). (3.36)

    Proof. We first show im(PX)=TXStn,k. Let XStn,k and VRn×k. One calculates

    X(PX(V))=X(VXXV+XπD+E(XV))=XVXV+πD+E(XV)=πD+E(XV).

    Moreover, using im(πD+E)=so(n), we obtain

    (PX(V))X=(VXXV+XπD+E(XV))X=VXVX+(πD+E(XV))=πD+E(XV).

    Hence X(PX(V))=πD+E(XV)=(PX(V))X follows, i.e. im(PX)TXStn,k as desired.

    We now assume VTXStn,k. By using XV=VX and πD|so(n)=idso(n), we calculate

    PX(V)=VXXV+XπD(XV)=VXXV+X(XV)=V

    proving PX|TXStn,k=idTXStn,k and implying that im(PX)TXStn,k is indeed an equality.

    It remains to show ker(PX)=(TXStn,k). Let VNXStn,k. We may write V=XΛ(D+E)1 with some suitable symmetric matrix ΛRk×ksym by exploiting Lemma 3.17. Consequently, we have

    PX(V)=PX(XΛ(D+E)1)=XΛ(D+E)1XX(XΛ(D+E)1)+XπD+E(XXΛ(D+E)1)=XπD+E(Λ(D+E)1)=0,

    by using Lemma 3.14, Claim 1 which shows πD+E(Λ(D+E)1)=0.

    Theorem 3.18 reproduces several results known in the literature.

    Remark 3.19. Let XStn,k. We obtain the following special cases for PX:Rn×kTXStn,k by using Corollary 3.16:

    1. For D=Ik and E=0 we get the formula

    PX(V)=VXXV+Xskew(XV)=(In12XX)V12XVX (3.37)

    that can be found for example in [2, Ex. 3.6.2] or [10, Eq. (2.4)]

    2. More generally, for D=2Ik and E=νIn with νR{2} one obtains

    PX(V)=VXXV+Xskew(XV)=V12XXV12XVX (3.38)

    reproducing the orthogonal projection from [14, Prop. 2].

    Next we determine an orthonormal basis of (TIn,kStn,k,,D,E()) which allows for computing the signature of ,D,E(), as well.

    Remark 3.20. We define the subsets B1,B2Rn×k such that B=B1B2 is an orthonormal basis of (TIn,kStn,k,,D,EIn,k). To this end, let EijRn×k denote the matrix whose entries fulfill (Eij)f=δifδj as usual. We set B1= for k=1 and define

    B1={1|sij|(EijEji)|sij=Dii+Eii+Djj+Ejj, 1i<jk},2kn. (3.39)

    Moreover, we set

    B2={1|Djj|Eij|k+1in,1jk},1k<n. (3.40)

    and B2= for k=n. A straightforward calculation shows V,WD,EIn,k=0 for all V,WB with VW. Moreover, for V=WB one obtains

    1|sij|(EijEji),1|sij|(EijEji)D,EIn,k=sij|sij|=±1,1i<jk (3.41)

    and

    1|Djj|Eij,1|Djj|EijD,EIn,k=Djj|Djj|=±1,k+1in, 1jk. (3.42)

    Hence B is in fact an orthonormal basis. Thus we may compute the signature of ,D,E(). The number of negative signs associated with ,D,E(), named index in [17, Chap. 2, Def. 18], is given by

    s={(i,j)1i<jk and sij<0}+(nk){j1jk and Djj<0}, (3.43)

    where S denotes the number of elements in the finite set S.

    Before we continue with the extrinsic approach, we briefly discuss the metric ,D,E() on Stn,k viewed as a pseudo-Riemannian reductive homogeneous SO(n)-space. This point of view allows for relating ,D,E() to the metrics investigated in [15]. For general properties of reductive homogeneous space we refer to [17, Chapter 11] as well as [18, Section 23.4].

    Throughout this subsection, we assume 1kn1 and n3. Then the Killing form on SO(n) given by

    ξ,η=(n2)tr(ξη),ξ,ηso(n)

    is negative definite, see e.g. [18, Sec. 21.6]. In addition, Stn,k is diffeomorphic to the reductive homogeneous space SO(n)/SO(nk), where SO(nk) is realized as a closed subgroup of SO(n) via

    SO(nk){[Ik00R]|RSO(nk)}SO(n)

    and a reductive split is given by so(n)=hm, where

    h={[000ξ22]|ξ22so(nk)} and m={[ξ11ξ21ξ210]|ξ11so(k), ξ21R(nk)×k},

    see e.g. [18, Sec. 23.5]. In particular, since SO(n)×Stn,k(R,X)RXStn,k is a transitive SO(n)-left action whose stabilizer subgroup of In,k coincides with SO(nk)SO(n), the map

    pr:SO(n)Stn,kSO(n)/SO(nk),RRIn,k (3.44)

    is a surjective submersion which induces a SO(n)-equivariant diffeomorphism

    ˇpr:SO(n)/SO(nk)Stn,k,RSO(nk)RIn,k. (3.45)

    Here RSO(nk)SO(n)/SO(nk) denotes the coset defined by RSO(n). We refer to [20, Thm. 6.4] and [21, Thm. 21.18] for more details on diffeomorphisms associated with transitive actions.

    In the sequel, we construct a scalar product

    ,red(D,E):so(n)×so(n)R

    on so(n) which induces a left-invariant metric on SO(n) such that (3.44) becomes a pseudo-Riemannian submersion. In addition, equipping SO(n)/SO(nk) with this submersion metric turns (3.45) into a SO(n)-equivariant isometry to (Stn,k,,D,E()).

    Throughout this section we denote by D,ERk×k diagonal matrices such that D and D+E are both invertible, see also Notation 3.13.

    Lemma 3.21. Let EijRn×n be the matrix whose entries fulfill (Eij)f=δifδj and let F=D+ERk×k. Then

    A:so(n)so(n),ξ=[ξ11ξ21ξ21ξ22]A(ξ)=[skew(ξ11(D+E))12Dξ2112ξ21Dξ22] (3.46)

    is linear, where ξ11so(k), ξ22so(nk) and ξ21R(nk)×k. Moreover, evaluating A at the basis {(EijEji)1i<jk} of so(n) yields

    A(EijEji)={Fii+Fjj2(EijEji)   if 1i<jk,12Djj(EijEji) if k+1in, 1jk,EijEji   if k+1i<jn. (3.47)

    In particular, A:so(n)so(n) as well as its restriction A|m:mm are linear isomorphisms.

    Proof. Clearly, A is linear. We show (3.47) by using the definition of A in (3.46). First we consider the case 1i<jk. Then EijEji is mapped by A to

    A(EijEji)=[skew((ˆEijˆEji)F)000]=[skew(ˆEijFjjˆEjiFii)000]=Fii+Fjj2(EijEji),

    with ˆEijRk×k defined by (ˆEij)f=δifδj. Next assume k+1in and 1jk. One obtains

    A(EijEji)=12Djj(EijEji).

    The equality A(EijEji)=EijEji for k+1i<jn is obvious.

    Lemma 3.22. Define

    ,red(D,E):so(n)×so(n)R,(ξ,η)ξ,ηred(D,E)=tr(ξA(η)), (3.48)

    where A:so(n)so(n) is the linear map from Lemma 3.21. Then the following assertions are fulfilled:

    1. ,red(D,E) is a scalar product on so(n).

    2. The restriction of ,red(D,E) to m defines a scalar product ,red(D,E):m×mR on m.

    3. Writing

    ξ=[ξ11ξ21ξ21ξ22]so(n)andη=[η11η21η21η22]so(n) (3.49)

    with ξ11,η11so(k) and ξ21,η21R(nk)×k yields

    ξ,ηred(D,E)=tr(ξ11skew(η11(D+E)))+tr(ξ21η21D)+tr(ξ22η22). (3.50)

    4. ,red(D,E) is Ad(SO(k))-invariant.

    5. Declaring Tepr:TeSO(n)Tpr(e)(SO(n)/SO(nk)) as an isometry defines a SO(n)-invariant pseudo-Riemannian metric on SO(n)/SO(nk) such that pr:SO(n)SO(n)/SO(nk) is a pseudo-Riemannian submersion, where SO(n) is equipped with the left-invariant metric defined by ,red(D,E).

    Proof. Obviously, ,red(D,E) is a bilinear form. Using the notation introduced in (3.49) one calculates

    ξ,ηred(D,E)=tr([ξ11ξ21ξ21ξ22][skew(η11(D+E))12Dη2112η21Dη22])=tr(ξ11(skew(η11(D+E)))+tr(ξ21η21D)+tr(ξ22η22)=η,ξred(D,E). (3.51)

    Hence ,red(D,E) is symmetric. Claim 3 follows by (3.51), as well. Moreover, ,red(D,E) is a scalar product since A:so(n)so(n) is a linear isomorphism by Lemma 3.21 showing Claim 1. Claim 2 follows since A|m:mm is an isomorphism, too.

    In order to show the Ad(SO(nk))-invariance we calculate

    Adg(ξ)=[Ik00R][ξ11ξ21ξ21ξ22][Ik00R]=[ξ11(Rξ21)Rξ21Rξ22R]

    for ξ=[x11ξ21ξ21ξ22]so(n) and g=[Ik00R]SO(nk)SO(n) implying

    Adg(ξ),Adg(η)red(D,E)=tr(ξ11(skew(η11(D+E))))+tr(((Rξ21)Rη21D)+tr((Rξ22R)Rη22R)=ξ,ηred(D,E)

    as desired.

    It remains to prove Claim 5. By (3.50) the vector spaces mso(n) and hso(n) are orthogonal complements with respect to ,red(D,E). Moreover, by exploiting the Ad(SO(nk))-invariance of ,red(D,E), this claim follows by [18, Prop. 23.23] which extends to the pseudo-Riemannian setting because its proof only relies on the non-degeneracy of the metric.

    After this preparation, we are in the position to show that ,red(D,E) has indeed the desired property. To this end, the tangent map of (3.44) at InSO(n) is determined as

    TInpr:so(n)TIn,kStn,k,ξξIn,k. (3.52)

    Proposition 3.23. Let SO(n)/SO(nk) be equipped with the pseudo-Riemannian metric constructed by means of the scalar product ,red(D,E):m×mR and let Stn,k be endowed with the metric ,D,E().

    1. The restriction of (3.52) to m, i.e. the linear map

    TInpr|m:mTIn,kStn,k,ξξIn,k (3.53)

    is an isometry, where TIn,kStn,k is equipped with the scalar product ,D,EIn,k.

    2. The SO(n)-equivariant diffeomorphism (3.45) is an isometry.

    Proof. We write ξ,ηm as

    ξ=[ξ11ξ21ξ210] andη=[η11η21η210]

    with ξ11,η11so(k) as well as ξ21,η21R(nk)×k and compute

    TInprξ,TInprηD,Epr(In)=[ξ11ξ21],[η11η21]D,EIn,k=tr(ξ11(skew(η11(D+E))))+tr(ξ21η21D)=ξ,ηred(D,E),

    where the last equality holds by Lemma 3.22, Claim 3. It remains to show Claim 2. Since the metric on SO(n)/SO(nk) induced by ,red(D,E) and the metric ,D,E() on Stn,k are both SO(n)-invariant, the map ˇpr:SO(n)/SO(nk)Stn,k is an isometry by Claim 1 due to its SO(n)-equivariance.

    Proposition 3.23 allows for relating the metric ,D,E()Γ(S2(TStn,k)) to the metrics on Stn,k defined in [15, Eq. (3.2)]. In order to compare these metrics we introduce some notation following [15]. We choose k1,,ksN with

    k1++ks=k and ki2 for all i{1,,s}

    and write

    D=diag(˜D11Ik1,,˜DssIks) and E=diag(˜E11Ik1,,˜EssIks), (3.54)

    where ˜Dii,˜EiiR. Using the notation from [15] with p=m and t=1 we rewrite ,red(D,E) as

    ,red(D,E)=si=1˜Dii+˜Eii+˜Dii+˜Eii2(n2),pi+1i<js˜Dii+˜Eii+˜Djj+˜Ejj2(n2),pij+si=1˜Dii2(n2),pi,s+1.

    Here ,pij=(n2)tr(()())|pij×pij denotes the Killing form on so(n) scaled by 1 restricted to pij. Hence ,red(D,E) coincides with the inner product defined in [15, Eq. (3.2)], where xi=xii and

    xij={˜Dii+˜Eii+˜Djj+˜Ejj2(n2) if 1ijs˜Dii2(n2)    if j=s+1 and 1is,

    provided that D and E are defined as in (3.54) as well as

    ˜Dii>0,i{1,,s} and ˜Dii+˜Eii+˜Djj+˜Ejj>0,i,j{1,,s}

    holds. This can be seen by observing that for ξm=p the unique decomposition of ξ into sums of ξijpij can be rewritten in terms of block matrices as

    Finally, we point out that the Einstein metrics discussed in [15, Sec. 6] yield the following equations for D and E

    x=˜Dii+˜Eiin2  for 1isy=˜Dii+˜Eii+˜Djj+˜Ejj2(n2)  for  1i<js,z=˜Dii2(n2)  for  1is,

    where x,y,z denote the parameters of the metric from [15, Eq. (6.2)]. Thus

    D=2(n2)zIkD+E=2z(n2)Ik+E=(n2)xIkE=(n2)(x2z)Ik

    and therefore y=(n2)(2z+(x2z)+2z+(x2z))2(n2)=x holds for x,zR. In particular, the metrics on Stn,k defined by ,D,E() contain only the two SO(n)×SO(k)-invariant Einstein metrics from [15], the so-called Jensen metrics. However, they do not contain the "new" Einstein metrics from that paper.

    Remark 3.24. Although the "new" Einstein metrics form [15] are not contained in the family of metrics on Stn,k defined by ,D,E(), we are not able to rule out that the family ,D,E() includes Einstein metrics different from the Jensen metrics. However, searching for Einstein metrics in ,D,E() is out of the scope of this text.

    The goal of this section is to derive an explicit expression for the spray SΓ(T(TStn,k)) associated with the metric ,D,E(). An expression for S yields an expression for the geodesic equation with respect to ,D,E() as an explicit second order ODE, as well.

    First we recall the definition of a metric spray, also known as spray associated with a metric, from [22, Chap. 8, §4] whose existence and uniqueness is proven in [22, Chap. 8, Thm. 4.2]. For general properties of sprays we refer to [22, Chap. 4, §3-4]. Moreover, a discussion of the relation of sprays to torsion-free covariant derivatives can be found in [22, Chap. 8 §2].

    Definition 4.1. Let (M,,) be a pseudo-Riemannian manifold. The metric spray SΓ(T(TM)) is the unique spray which is associated with the Levi-Civita covariant derivative defined by the pseudo-Riemannian metric ,.

    An expression of a metric spray in local coordinates is given in (4.2) below. Next we discuss the relation of metric sprays to Lagrangian mechanics.

    Let (M,,) be pseudo-Riemannian and let ω0Γ(Λ2T(TM)) denote the canonical symplectic form on TM. It is given by

    ω0=d θ0

    with θ0Γ(T(TM)) being the canonical 1-form on TM. We refer to [16, Sec. 6.2] for the definition of ω0 and θ0. Consider the Lagrange function

    L:TMR,vxL(vx)=12vx,vxx.

    Let FL:TMTM denote the fiber derivative of L defined by

    ((FL)(vx))(wx)=d d tL(vx+twx)|t=0,xM,vx,wxTxM,

    see e.g. [16, Eq. (7.2.1)]. The pullback

    ωL=(FL)ω0

    is a closed 2-from on TM, the so-called Lagrangian 2-form, see [16, Sec. 7.2]. In addition, ωL is non-degenerated, i.e. symplectic, since FL:TMTM is a diffeomorphism due to

    FL:TMTM,vxFL(vx)=vx, (4.1)

    by [16, Eq. (7.5.3)]. Moreover, the energy

    EL:TMR,vx((FL)(vx))(vx)L(vx)

    associated with L fulfills EL=L, see e.g. [16, Sec. 7.3]. Let XELΓ(T(TM)) denote the Lagrangian vector field and write iXELωL for the insertion of XEL into the first argument of ωL as usual. Then XEL is uniquely determined by

    iXELωL=d ELωL(XEL,V)=d EL(V) for all VΓ(T(TM)).

    according to [16, Sec. 7.3]. Moreover, the Lagrangian vector field XEL coincides with the spray associated with the metric ,, see e.g. [16, Sec. 7.5]. It is exactly the so-called canonical spray from [22, Chap. 7, §7] which coincides with the metric spray, see [22, Chap. 8, Thm. 4.2]. Finally, we mention a local expression for sprays, see e.g. [22, Chap. 8, §4]. A metric spray S:TMT(TM) can be represented in a chart (TU,(x,v)) of TM induced by a chart (U,x) of M by

    S(x,v)=(x,v,v,Γx(v,v)). (4.2)

    Here Γx denotes the quadratic map defined by (Γx(v,v))k=Γkij(x)vivj using Einstein summation convention, where Γkij are the Christoffel symbols of the Levi-Civita covariant derivative with respect to the chart (U,x). In order to apply these general results to our particular situation, we introduce some notation.

    Notation 4.2. Throughout this section URn×k denotes an open subset of Rn×k with the property from Lemma 3.8. Moreover, we denote by ˜L the Lagrange function

    ˜L:TUR,(X,V)˜L(X,V)=12V,VD,EX, (4.3)

    where we identify TUU×Rn×k as usual.

    We use the formula for ω0Γ(Λ2T(TU)) on TU given in the next remark.

    Remark 4.3. The canonical symplectic form ω0Γ(Λ2T(TU)) on TU is given by

    ω0|(X,V)((X,V,Y,Z),(X,V,˜Y,˜Z))=tr(Y˜Z)tr(˜YZ), (4.4)

    for (X,V,Y,Z),(X,V,˜Y,˜Z)T(TU) identifying T(TU)U×(Rn×k)×Rn×k×(Rn×k) as well as Rn×k(Rn×k) via Vtr(V()). Indeed, Equation (4.4) follows by the local formula for the canonical symplectic form ω0 on TU, see e.g. [16, Sec. 6.2], applied to the gobal chart (U,idU)=(U,Xij).

    We now calculate the Lagrangian 2-from ω˜L=(F˜L)ω0. To this end, we first determine the fiber derivative F˜L:TUTU and its tangent map.

    Lemma 4.4. For (X,V)TU the fiber derivative F˜L:TUTU of ˜L is given by

    F˜L(X,V)=(X,tr((VD+XXVE)())). (4.5)

    Proof. Let (X,V),(X,W)TU. We have (F˜L(X,V))(X,W)=V,WD,EX by the Definition of ˜L and (4.1). Using the definition of ,D,E() and exploiting properties of the trace we obtain

    (F˜L(X,V))(X,W)=tr(VWD)+tr(VXXWE)=tr((VD+XXVE)W)

    as desired.

    Lemma 4.5. The tangent map T(F˜L):T(TU)T(TU) is given by

    (T(F˜L))(X,V,Y,Z)=(F˜L(X,V),Y,tr((ZD+YXVE+XYVE+XXZE)()))

    for (X,V,Y,Z)T(TU)U×(Rn×k)3, where we identify T(TU)U×(Rn×k)×Rn×k×(Rn×k).

    Proof. Let (X,V,Y,Z)T(TU). The smooth curve γ:(ϵ,ϵ)t(X+tY,V+tZ)TU, for ϵ>0 sufficiently small, fulfills γ(0)=(X,V) with ˙γ(0)=(Y,Z). Then

    d d tF˜L(γ(t))|t=0=(Y,tr((ZD+YXVE+XYVE+XXZE)())).

    This yields the desired result.

    Lemma 4.6. The Lagrangian 2-form ω˜L=(F˜L)ω0Γ(Λ2T(TU)) is given by

    ω˜L|(X,V)((X,V,Y,Z),(X,V,˜Y,˜Z)=tr(Y(˜ZD+˜YXVE+X˜YVE+XX˜ZE))tr(˜Y(ZD+YXVE+XYVE+XXZE)) (4.6)

    with (X,V)TUU×Rn×k and (X,V,Y,Z),(X,V,˜Y,˜Z)T(X,V)TU.

    Proof. Using the formula for ω0Γ(Λ2T(TU)) from Remark 4.3, a straightforward calculation shows that ω˜L=(F˜L)ω0 is given by (4.6). To this end, the formulas from Lemma 4.4 and Lemma 4.5 are plugged into the definition of the pull-back (F˜L)ω0.

    Next the spray ˜SΓ(T(TU)) associated with ,D,E() is calculated exploiting ˜S=XE˜L, where XE˜L is the Lagrangian vector field. A closed form expression for ˜S(X,V) is obtained for all (X,V)Stn,k×Rn×kTU.

    Lemma 4.7. For (X,V)TU and (X,V,Y,Z)T(X,V)TU one has

    d E˜L|(X,V)(X,V,Y,Z)=tr(VZD)+tr(ZXXVE)+tr(VYXVE). (4.7)

    Proof. Let (X,V),(Y,Z)TU. We calculate

    d d tE˜L(X+tY,V+tZ)|t=0=12(tr(ZVD+VZD)+tr(ZXXVE+VYXVE+VXYVE+VXXZE)).

    Using properties of the trace yields the desired result.

    Next we consider a linear matrix equation of a certain form. We need to solve this equation for computing the metric spray on TU, see Proposition 4.9. Moreover, one encounters this equation in the proof of Proposition 5.2 on pseudo-Riemannian gradients below.

    Lemma 4.8. Let D,ERk×kdiag such that D and D+E are both invertible and let WRn×k. Moreover, let URn×k be open with the property from Lemma 3.8. Then for XU the linear equation

    ˜ΓD+XX˜ΓE=W (4.8)

    has a unique solution in terms of ˜Γ. Moreover, for XStn,k, it is explicitly given by

    ˜Γ=(WXXW(D+E)1E)D1. (4.9)

    Proof. For each XU the linear map ϕ:Rn×k˜Γ˜ΓD+XX˜ΓERn×k is an isomorphism since the bilinear form

    Rn×k×Rn×kR,(Y,Z)tr(Vϕ(W))=V,WD,EX

    is non-degenerated by assumption. Hence (4.8) admits a unique solution. Now assume XStn,k. We briefly explain how (4.9) can be derived. By exploiting XX=Ik, Equation (4.8) implies

    XW=X˜ΓD+X˜ΓE=X˜Γ(D+E)X˜Γ=XW(D+E)1.

    Plugging X˜Γ=XW(D+E)1 into (4.8) yields

    ˜ΓD+X(XW(D+E)1)E=W˜Γ=(WX(XW(D+E)1)E)D1.

    A straightforward calculation shows that ˜Γ is indeed a solution of (4.8).

    Proposition 4.9. The spray ˜SΓ(T(TU)) associated with the metric ,D,E() is given by

    ˜S(X,V)=(X,V,V,˜Γ)=(X,V,V,˜ΓX(V,V)) (4.10)

    for all (X,V)TUU×Rn×k. Here ˜Γ=˜ΓX(V,V)Rn×k depending on (X,V)TU is the unique solution of the linear equation

    ˜ΓD+XX˜ΓE=VXVE+XVVEVEVX (4.11)

    in terms of \widetilde{\Gamma} with fixed (X, V) \in TU . Moreover, for (X, V) \in \mathrm{St}_{{{n}}, {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}} one has

    \begin{equation} \begin{split} \widetilde{\Gamma}_X(V, V) & = \big( V X^{\top} V E + X V^{\top} V E - V E V^{\top} X \big) D^{-1} \\ &\quad + \big( X X^{\top} V E V^{\top} X - X (X^{\top} V)^2 E - X V^{\top} V E \big)(D + E)^{-1} E D^{-1} . \end{split} \end{equation} (4.12)

    Proof. Using \widetilde{S} = X_{E_{\widetilde{L}}} we compute \widetilde{S} via solving i_{X_{E_{\widetilde{L}}}} \omega_{\widetilde{L}} = {\rm{d}}\ E_{\widetilde{L}} for X_{E_{\widetilde{L}}} , i.e. \widetilde{S} = X_{E_{\widetilde{L}}} fulfills

    \begin{equation} \omega_{\widetilde{L}}\big(X_{E_{\widetilde{L}}}(X, V), (X, V, Y, Z) \big) = {\rm{d}}\ E_{\widetilde{L}}\big\vert_{{(X, V)}}(X, V, Y, Z) . \end{equation} (4.13)

    for all (X, V, Y, Z) \in T(T U) . Since \omega_{\widetilde{L}} is non-degenerated, X_{E_{\widetilde{L}}} is uniquely determined by (4.13). The local form of a metric spray, see (4.2), motivates the Ansatz

    \begin{equation*} X_{E_{\widetilde{L}}}(X, V) = \big(X, V, V, - \widetilde{\Gamma}_X(V, V) \big) = \big(X, V, V, - \widetilde{\Gamma} \big) \end{equation*}

    with \widetilde{\Gamma} = \widetilde{\Gamma}_X(V, V) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} depending on (X, V) \in T U . Inserting X_{E_{\widetilde{L}}} into \omega_{\widetilde{L}} from Lemma 4.6 yields the 1 -form

    \begin{equation} \begin{split} (i_{X_{E_{\widetilde{L}}}} \omega_{\widetilde{L}} )\big\vert_{{(X, V)}}(X, V, Y, Z) & = \omega_{\widetilde{L}}\big\vert_{{(X, V)}}\big( X_{E_{\widetilde{L}}}(X, V), (X, V, Y, Z) \big) \\ & = {\rm{tr}}\Big( V^{\top} \Big( Z D + Y X^{\top} V E + X Y^{\top} V E + X X^{\top} Z E \Big) \\ &\quad -{\rm{tr}}\Big( Y^{\top} \Big( - \widetilde{\Gamma} D + V X^{\top} V E + X V^{\top} V E - X X^{\top} \widetilde{\Gamma} E \Big) \Big) \end{split} \end{equation} (4.14)

    with (X, V) \in TU and (X, V, Y, Z) \in T(TU) . Using (4.14) and the formula for {\rm{d}}\ E_{\widetilde{L}} from Lemma 4.7, the equation i_{X_{E_{\widetilde{L}}}} \omega_{\widetilde{L}} = {\rm{d}}\ E_{\widetilde{L}} becomes

    \begin{equation} \begin{split} &{\rm{tr}}(V^{\top} Z D) + {\rm{tr}}(Z^{\top} X X^{\top} V E) + {\rm{tr}}(V^{\top} Y X^{\top} V E) \\ &\quad = {\rm{tr}}\Big( V^{\top} \Big( Z D + Y X^{\top} V E + X Y^{\top} V E + X X^{\top} Z E \Big) \\ &\quad\quad -{\rm{tr}}\Big( Y^{\top} \Big( - \widetilde{\Gamma} D + V X^{\top} V E + X V^{\top} V E - X X^{\top} \widetilde{\Gamma} E \Big) \Big) \end{split} \end{equation} (4.15)

    for all (X, V, Y, Z) \in TU . Clearly, Equation (4.15) is equivalent to

    \begin{equation*} {\rm{tr}}(Y^{\top} (V E V^{\top} X)) = {\rm{tr}}\big( Y^{\top} ( - \widetilde{\Gamma} D + V X^{\top} V E + X V^{\top} V E - X X^{\top} \widetilde{\Gamma} E ) \big) \end{equation*}

    for all Y \in \mathbb{{{R}}}^{n \times k} . This can be equivalently rewritten as

    \begin{equation} \widetilde{\Gamma} D + X X^{\top} \widetilde{\Gamma} E = V X^{\top} V E + X V^{\top} V E - V E V^{\top} X \end{equation} (4.16)

    showing the first claim.

    We now assume X \in \mathrm{St}_{{{n}}, {{k}}} . Writing W = V X^{\top} V E + X V^{\top} V E - V E V^{\top} X and invoking Lemma 4.8 in order to solve (4.16) for \widetilde{\Gamma} yields

    \begin{equation*} \begin{split} \widetilde{\Gamma} & = W D^{-1} - X X^{\top} W (D + E)^{-1} E D^{-1} \\ & = \big( V X^{\top} V E + X V^{\top} V E - V E V^{\top} X \big) D^{-1} \\ &\quad + \big( X X^{\top} V E V^{\top} X - X (X^{\top} V)^2 E - X V^{\top} V E \big)(D + E)^{-1} E D^{-1} \end{split} \end{equation*}

    as desired.

    Remark 4.10. Obviously, for E = 0 , Proposition 4.9 implies \widetilde{\Gamma}_X(V, V) = 0 for all (X, V) \in TU .

    Proposition 4.9 admits a relatively simple expression for \widetilde{S} \! \in \Gamma^{\infty}\big(T (TU)\big) evaluated at (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} for a subfamily of \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} . Since this subfamily will be discussed several times below, it deserves its own notation.

    Notation 4.11. We write \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} for the covariant 2 -tensor \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} which is obtained by specifying E = \nu I_k with \nu \in \mathbb{{{R}}} , i.e.

    \begin{equation*} \big\langle V, W \big\rangle^{D, \nu}_X = {\rm{tr}}\big(V^{\top} W D \big) + \nu {\rm{tr}}\big( V^{\top} X X^{\top} W \big), \quad X \in \mathbb{{{R}}}^{{{n}} \times {{k}}} \ {\text{and}}\ V, W \in T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}} . \end{equation*}

    Unless indicated otherwise, pull-backs of \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathbb{{{R}}}^{{{n}} \times {{k}}}) \big) to submanifolds of \mathbb{{{R}}}^{{{n}} \times {{k}}} are omitted in the notation. Moreover, we assume that D and \nu are chosen such that \mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} \big) is a pseudo-Riemannian submanifold. In particular, we assume that D and D + \nu I_k are both invertible.

    Corollary 4.12. The spray \widetilde{S} \in \Gamma^{\infty}\big(T (T U) \big) on TU associated with \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* U)\big) evaluated at (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} is given by

    \begin{equation} \widetilde{S}(X, V) = \big(X, V, V, - \widetilde{\Gamma}_X(V, V) \big), \end{equation} (4.17)

    where

    \begin{equation} \widetilde{\Gamma}_X(V, V) = \Big( 2 \nu V X^{\top} V + \nu X V^{\top} V \big( D \big(D + \nu I_k \big)^{-1} \big) - 2 \nu^2 X (X^{\top} V)^2 (D + \nu I_k)^{-1} \Big) D^{-1} . \end{equation} (4.18)

    Proof. Let (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} and write \widetilde{\Gamma} = \widetilde{\Gamma}_X(V, V) for short. Plugging E = \nu I_k into Formula (4.12) from Proposition 4.9 and using X^{\top} V = - V^{\top} X we obtain

    \begin{equation*} \begin{split} \widetilde{\Gamma} & = \Big( V X^{\top} V E + X V^{\top} V E - V E V^{\top} X \Big) D^{-1} \\ &\quad + \Big( X X^{\top} V E V^{\top} X - X (X^{\top} V)^2 E - X V^{\top} V E \Big)(D + E)^{-1} E D^{-1} \\ & = \nu \Big( V X^{\top} V + X V^{\top} V - V V^{\top} X \Big) D^{-1} + \nu^2 \Big( X X^{\top} V V^{\top} X - X (X^{\top} V)^2 - X V^{\top} V \Big)(D + \nu I_k)^{-1} D^{-1} \\ & = \nu \Big( V X^{\top} V + X V^{\top} V + V X^{\top} V \Big) D^{-1} + \nu^2 \Big( - X (X^{\top} V)^2 - X (X^{\top} V)^2 - X V^{\top} V \Big)(D + \nu I_k)^{-1} D^{-1} \\ & = \Big( 2 \nu V X^{\top} V + X V^{\top} V \big(\nu I_k \big) - 2 \nu^2 X (X^{\top} V)^2 \big(D + \nu I_k \big)^{-1} - X V^{\top} V \big( \nu^2 (D + \nu I_k)^{-1} \big) \Big) D^{-1} \\ & = \Big( 2 \nu V X^{\top} V + X V^{\top} V \big( \nu I_k - \nu^2 (D + \nu I_k)^{-1} \big) - 2 \nu^2 X (X^{\top} V)^2 (D + \nu I_k)^{-1} \Big) D^{-1} \\ & = \Big( 2 \nu V X^{\top} V + \nu X V^{\top} V \big( D \big(D + \nu I_k \big)^{-1} \big) - 2 \nu^2 X (X^{\top} V)^2 (D + \nu I_k)^{-1} \Big) D^{-1} , \end{split} \end{equation*}

    where the last equality holds due to

    \begin{equation*} \big( \nu I_k - \nu^2 (D + \nu I_k)^{-1}\big)_{ii} = \nu - \tfrac{\nu^2}{D_{ii} + \nu} = \tfrac{\nu(D_{ii} + \nu) - \nu^2}{D_{ii} + \nu} = \nu (D \big(D + \nu I_k \big)^{-1})_{ii} \end{equation*}

    for i \in \{1, \ldots, k\} .

    We now determine the spray S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}})\big) associated with the metric \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} . To this end, a result from [16, Prop. 8.4.1] is exploited which is stated for Riemannian manifolds. The proof works for pseudo-Riemannian manifolds, as well, since it only exploits the non-degeneracy of the metric. We reformulate it in the following proposition.

    Proposition 4.13. Let M \subseteq \widetilde{M} be a pseudo-Riemannian submanifold of a pseudo-Riemannian manifold \big(\widetilde{M}, \langle \cdot, \cdot \rangle\big) and let \widetilde{S} \in \Gamma^{\infty}\big(T (T\widetilde{M}) \big) denote the metric spray on T\widetilde{M} . Then the spray S \in \Gamma^{\infty}\big(T (TM) \big) on TM associated with the induced pseudo-Riemannian metric is given by

    \begin{equation} S = T P \circ \widetilde{S} \big\vert_{{TM}}\colon TM \to T(TM), \end{equation} (4.19)

    where P \colon T\widetilde{M}\big\vert_{{M}}\to TM denotes the vector bundle morphism that is defined fiber-wise by the orthogonal projections P_x \colon T_x \widetilde{M} \to T_x M \subseteq T_x \widetilde{M} with respect to \langle \cdot, \cdot \rangle , where x \in M .

    Lemma 4.14. The tangent map T P \colon T (\mathrm{St}_{{{n}}, {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}}) \to T(T \mathrm{St}_{{{n}}, {{k}}}) of

    \begin{equation} P \colon \mathrm{St}_{{{n}}, {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}} \to T \mathrm{St}_{{{n}}, {{k}}}, \quad (X, V) \mapsto (X, P_X(V) ), \end{equation} (4.20)

    where P_X(V) = V - X X^{\top} V + X \pi^{D + E}(X^{\top} V) is the orthogonal projection from Theorem 3.18, is given by

    \begin{equation} T P(X, V, Y, Z) = \big( X, V, Y, Z - X Y ^{\top} V - X X^{\top} Z + X \pi^{D + E}(Y^{\top} V + X^{\top} Z) )\big) \end{equation} (4.21)

    for all (X, V, Y, Z) \in T(\mathrm{St}_{{{n}}, {{k}}} \times \mathbb{{{R}}}^{n \times k}) \cong T \mathrm{St}_{{{n}}, {{k}}} \times (\mathbb{{{R}}}^{n \times k})^2 .

    Proof. By exploiting \pi^{D + E}(X^{\top} V) = X^{\top} V due to X^{\top} V = - V^{\top} X \in \mathfrak{{{so}}}(k) for (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} one calculates

    \begin{equation*} \begin{split} T_{(X, V)} P( Y, Z) & = \big( Y, Z - Y X^{\top} V - X Y^{\top} V - X X^{\top} Z + Y \pi^{D + E}(X^{\top} V) + X \pi^{D + E}(Y^{\top} V + X^{\top} Z) \big) \\ & = \big( Y, Z - X Y ^{\top} V - X X^{\top} Z + X \pi^{D + E}(Y^{\top} V + X^{\top} Z) \big) , \end{split} \end{equation*}

    where (X, V, Y, Z) \in T(\mathrm{St}_{{{n}}, {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}}) .

    Theorem 4.15. The spray S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}})\big) associated with \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} is given by

    \begin{equation} S(X, V) = \big(X, V, V, -\widetilde{\Gamma}_X(V, V) - X V^{\top} V + X X^{\top} \widetilde{\Gamma}_X(V, V) + X \pi^{D + E}\big(V^{\top} V - X^{\top} \widetilde{\Gamma}_X(V, V) \big) \big) \end{equation} (4.22)

    for all (X, V) \in T\mathrm{St}_{{{n}}, {{k}}} . Here \widetilde{\Gamma}_X(V, V) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} depending on (X, V) \in T\mathrm{St}_{{{n}}, {{k}}} is given by

    \begin{equation} \begin{split} \widetilde{\Gamma}_X(V, V) & = \big( V X^{\top} V E + X V^{\top} V E - V E V^{\top} X \big) D^{-1} \\ &\quad + \big( X X^{\top} V E V^{\top} X - X (X^{\top} V)^2 E - X V^{\top} V E \big)(D + E)^{-1} E D^{-1} . \end{split} \end{equation} (4.23)

    Proof. One can view \mathrm{St}_{{{n}}, {{k}}} equipped with \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} as a pseudo-Riemannian submanifold of \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big) according to Lemma 3.8. Let \widetilde{S} \in \Gamma^{\infty}\big(T (TU)\big) be the metric spray on TU determined in Proposition 4.9. Then S = TP \circ \widetilde{S}\big\vert_{{T \mathrm{St}_{{{n}}, {{k}}}}} holds by Proposition 4.13. Using Lemma 4.14 yields

    \begin{equation*} \begin{split} S(X, V) & = TP \circ \widetilde{S} \big\vert_{{T \mathrm{St}_{{{n}}, {{k}}}}}(X, V) = TP \big(X, V, V, - \widetilde{\Gamma}_X(V, V) \big) \\ & = \big(X, V, V, - \widetilde{\Gamma}_X(V, V) - X V ^{\top} V + X X^{\top} \widetilde{\Gamma}_X(V, V) + X \pi^{D + E}\big(V^{\top} V - X^{\top} \widetilde{\Gamma}_X(V, V) \big)\big) \end{split} \end{equation*}

    for all (X, V) \in T\mathrm{St}_{{{n}}, {{k}}} as desired.

    Remark 4.16. We often denote the spray on T\mathrm{St}_{{{n}}, {{k}}} associated with \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} from Theorem 4.15 by

    \begin{equation*} S(X, V) = \big(X, V, V, - \Gamma \big) = \big(X, V, V, - \Gamma_X(V, V) \big), \end{equation*}

    i.e. we write - \Gamma or - \Gamma_X(V, V) for the fourth component of S . For (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} it is given by

    \begin{equation} - \Gamma_X(V, V) = -\widetilde{\Gamma}_X(V, V) - X V^{\top} V + X X^{\top} \widetilde{\Gamma}_X(V, V) + X \pi^{D + E}\big(V^{\top} V - X^{\top} \widetilde{\Gamma}_X(V, V) \big) \end{equation} (4.24)

    according to Theorem 4.15, where \widetilde{\Gamma}_X(V, V) is determined by (4.23). Obviously, Equation (4.24) yields a well-defined expression for all X \in \mathbb{{{R}}}^{{{n}} \times {{k}}} and V \in \mathbb{{{R}}}^{{{n}} \times {{k}}} which is quadratic in V . Hence, by polarization, (4.24) can be viewed as the definition of the smooth map

    \begin{equation} \Gamma \colon U \to \mathrm{S}^2\big( (\mathbb{{{R}}}^{{{n}} \times {{k}}})^*\big) \otimes \mathbb{{{R}}}^{{{n}} \times {{k}}}, \quad X \mapsto \big( (V, W) \mapsto \Gamma_X(V, W) \big) . \end{equation} (4.25)

    Clearly, Equation (4.25) yields a smooth extension of the fourth component of the metric spray S \! \in \! \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}})\big) . This extension is used in Proposition 6.5 and Proposition 6.8 below.

    Corollary 4.17. The spray S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}}) \big) associated with the metric \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} from Theorem 4.15 has the following properties:

    1. The spray S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}}) \big) is complete.

    2. The maximal integral curve \mathbb{{{R}}} \ni t \mapsto \Phi^S_t\big((X_0, V_0) \big) = \big(X(t), V(t) \big) \in T \mathrm{St}_{{{n}}, {{k}}} of S through the point (X_0, V_0) \in T \mathrm{St}_{{{n}}, {{k}}} at t = 0 fulfills the explicit non-linear first order ODE

    \begin{equation} \begin{split} \dot{X} & = V \\ \dot{V} & = -\widetilde{\Gamma}_X(V, V) - X V^{\top} V + X X^{\top} \widetilde{\Gamma}_X(V, V) + X \pi^{D + E}\big(V^{\top} V - X^{\top} \widetilde{\Gamma}_X(V, V) \big), \end{split} \end{equation} (4.26)

    with initial condition \big(X(0), V(0) \big) = \big(X_0, V_0 \big) \in T \mathrm{St}_{{{n}}, {{k}}} writing X = X(t) and V = V(t) for short.

    3. Let {\rm{pr}} \colon T \mathrm{St}_{{{n}}, {{k}}} \to \mathrm{St}_{{{n}}, {{k}}} be the canonical projection. The curve \mathbb{{{R}}} \! \ni \! t \! \mapsto \! {\rm{pr}} \! \circ \Phi^S_t(X_0, V_0) \! = \! X(t) \! \in \mathrm{St}_{{{n}}, {{k}}} is a geodesic with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} through the point X(0) = X_0 \in \mathrm{St}_{{{n}}, {{k}}} with initial velocity \dot{X}(0) = V_0 \in T_{X_0} \mathrm{St}_{{{n}}, {{k}}} .

    4. The geodesic equation on \mathrm{St}_{{{n}}, {{k}}} with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} is given by the non-linear explicit second order ODE

    \begin{equation} \ddot{X} = -\widetilde{\Gamma}_X(\dot{X}, \dot{X}) - X \dot{X}^{\top} \dot{X} + X X^{\top} \widetilde{\Gamma}_X(\dot{X}, \dot{X}) + X \pi^{D + E}(\dot{X}^{\top} \dot{X} - X^{\top} \widetilde{\Gamma}_X(\dot{X}, \dot{X}) \big) \end{equation} (4.27)

    with initial conditions X(0) = X_0 \in \mathrm{St}_{{{n}}, {{k}}} and \dot{X}(0) = \dot{X}_0 \in T_{X_0} \mathrm{St}_{{{n}}, {{k}}} .

    Proof. We first show that S is complete. The transitive O(n) -action \Psi acts on \big(\mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) by isometries according to Lemma 3.1, i.e. \big(\mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big) is a compact pseudo-Riemannian homogeneous manifold. Hence completeness follows by [23].

    The other statements are well-known consequences of general properties of sprays associated with a metric, see e.g. [16, Sec. 7.5], combined with the explicit formula for S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}}) \big) from Theorem 4.15.

    The formula for the metric spray S from Theorem 4.15 admits a simplification for \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} .

    Corollary 4.18. For \big(\mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} \big) the metric spray is given by S(X, V) = \big(X, V, V, - \Gamma_X(V, V)\big) with

    \begin{equation} -\Gamma_X(V, V) = 2 \nu V V^{\top} X D^{-1} + 2 \nu X (X^{\top} V)^2 D^{-1} - X V^{\top} V + X \pi^{D + \nu I_k}(V^{\top} V) \end{equation} (4.28)

    for (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} . Moreover, the geodesic equation reads

    \begin{equation} \ddot{X} = 2 \nu \dot{X} \dot{X}^{\top} X D^{-1} + 2 \nu X (X^{\top} \dot{X})^2 D^{-1} - X \dot{X}^{\top} \dot{X} + X \pi^{D + \nu I_k}(\dot{X}^{\top} \dot{X}) . \end{equation} (4.29)

    Proof. Let (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} . Using the formula for \widetilde{\Gamma}_X(V, V) from Corollary 4.12 we calculate

    \begin{equation*} \begin{split} X^{\top} \widetilde{\Gamma}_X(V, V) & = X^{\top} \Big( 2 \nu V X^{\top} V + \nu X V^{\top} V \big( D \big(D + \nu I_k \big)^{-1} \big) - 2 \nu^2 X (X^{\top} V)^2 (D + \nu I_k)^{-1} \Big) D^{-1} \\ & = 2 \nu X^{\top} V X^{\top} V D^{-1} + \nu V^{\top} V \big(\big(D + \nu I_k \big)^{-1} D \big) D^{-1} - 2 \nu^2 (X^{\top} V)^2 (D + \nu I_k)^{-1} D^{-1} \\ & = 2 \nu (X^{\top} V)^2 D^{-1} - 2 \nu^2 (X^{\top} V)^2 (D + \nu I_k)^{-1} D^{-1} + \nu V^{\top} V \big(\big(D + \nu I_k \big)^{-1} \big) \\ & = (X^{\top} V)^2 \Big( 2 \nu D^{-1} -2 \nu^2 (D + \nu I_k)^{-1} D^{-1} \Big) + \nu V^{\top} V \big(\big(D + \nu I_k \big)^{-1} \big) \\ & = 2 \nu (X^{\top} V)^2 (D + \nu I_k)^{-1} + \nu V^{\top} V (D + \nu I_k)^{-1} \\ & = \nu \Big( V^{\top} V + 2 (X^{\top} V)^2\Big) (D + \nu I_k)^{-1} , \end{split} \end{equation*}

    where the identity

    \begin{equation*} \big(2 \nu D^{-1} - 2 \nu^2 (D + \nu I_k)^{-1} D^{-1} \big)_{ii} = \tfrac{2 (\nu D_{ii} + \nu^2) - 2 \nu^2}{(D_{ii} + \nu) D_{ii} } = \tfrac{2 \nu}{D_{ii} + \nu} = 2 \nu \big((D + \nu I_k)^{-1} \big)_{ii} \end{equation*}

    is used. This yields

    \begin{equation*} X X^{\top} \widetilde{\Gamma}_X(V, V) = \nu X \Big( V^{\top} V + 2 (X^{\top} V)^2\Big) (D + \nu I_k)^{-1} . \end{equation*}

    Moreover, using the symmetry of \nu\big(V^{\top} V + 2 (X^{\top} V)^2 \big) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}} we obtain by Lemma 3.14, Claim 1

    \begin{equation*} \pi^{D + \nu I_k}\big(X^{\top} \widetilde{\Gamma}_X(V, V) \big) = \pi^{D + \nu I_k}\big( \nu \big( V^{\top} V + 2 (X^{\top} V)^2\big) (D + \nu I_k)^{-1} \big) = 0 . \end{equation*}

    Therefore \Gamma_X(V, V) can be obtained by Theorem 4.15 via calculating

    \begin{equation*} \begin{split} - \Gamma_X(V, V) & = -\widetilde{\Gamma}_X(V, V) - X V^{\top} V + X X^{\top} \widetilde{\Gamma}_X(V, V) + X \pi^{D + \nu I_k}\big(V^{\top} V - X^{\top} \widetilde{\Gamma}_X(V, v) \big) \\ & = \Big( - 2 \nu V X^{\top} V D^{-1} - \nu X V^{\top} V \big(D + \nu I_k \big)^{-1} + 2 \nu^2 X (X^{\top} V)^2 (D + \nu I_k)^{-1} D^{-1} \Big) - X V^{\top} V \\ &\quad + \Big( \nu X V^{\top} V (D + \nu I_k)^{-1} + 2 \nu X (X^{\top} V)^2 (D + \nu I_k)^{-1} \Big) + X \pi^{D + \nu I_k}(V^{\top} V) \\ & = 2 \nu V V^{\top} X D^{-1} + 2 X (X^{\top} V)^2 (D + \nu I_k)^{-1} (\nu^2 D^{-1} + \nu I_k) - X V^{\top} V + X \pi^{D + \nu I_k}(V^{\top} V) \\ & = 2 \nu V V^{\top} X D^{-1} + 2 \nu X (X^{\top} V)^2 D^{-1} - X V^{\top} V + X \pi^{D + \nu I_k}(V^{\top} V) , \end{split} \end{equation*}

    where the last equality follows due to

    \begin{equation*} \big((D + \nu I_k)^{-1} (\nu^2 D^{-1} + \nu I_k)\big)_{ii} = \tfrac{ (\nu^2 / D_{ii} ) + \nu}{D_{ii} + \nu} = \tfrac{\nu (\nu + D_{ii})}{ D_{ii} (\nu + D_{ii})} = \nu \big( D^{-1} \big)_{ii}. \end{equation*}

    This yields the desired result.

    Remark 4.19. Corollary 4.18 generalizes the geodesic equation from [13]. Indeed, setting D = 2 I_k and \nu = - \frac{2 \alpha + 1}{\alpha + 1} with \alpha \in \mathbb{{{R}}}\setminus \{-1\} yields

    \begin{equation} - \Gamma_X(V, V) = \nu V V^{\top} X + \nu X (X^{\top} V)^2 - X V^{\top} V \end{equation} (4.30)

    due to \pi^{(2 + \nu) I_k}(V^{\top} V) = {\rm{skew}}(V^{\top} V) = 0 in accordance with [13, Eq. (65)].

    Remark 4.20. We are not aware of an explicit solution of the geodesic equation for general diagonal matrices D and E . To our best knowledge, an explicit solution is only known for the special case D = 2 I_k and E = \nu I_k , see [13]. Nevertheless, one could exploit that \big(T\mathrm{St}_{{{n}}, {{k}}}, \omega_{(T\iota)^* \widetilde{L}}, (T\iota)^* \widetilde{L}\big) defines a Hamiltonian system whose Hamiltonian vector field is given by the metric spray S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}}) \big) . This point of view would allow to study the geodesic equation using the theory of integrable systems. However, investigating these aspects in detail is out of the scope of this paper. In this context, we only refer to [24], where geodesic flows on the cotangent bundle T^* \mathrm{St}_{{{n}}, {{k}}} and their integrability are studied.

    We now determine pseudo-Riemannian gradients and pseudo-Riemannian Hessians of smooth functions on \mathrm{St}_{{{n}}, {{k}}} . Specific results from [14] are generalized, where similar ideas were used to obtain the gradients and Hessians of smooth function on \mathrm{St}_{{{n}}, {{k}}} with respect to the one-parameter family of metrics from [13]. Moreover, similar formulas for gradients and Hessians on \mathrm{St}_{{{n}}, {{k}}} with respect to a family of metrics corresponding to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} , where D = \alpha_0 I_k and E = (\alpha_1 - \alpha_0) I_k with \alpha_0, \alpha_1 \in \mathbb{{{R}}} , i.e. a scaled version of the metrics introduced in [13], are independently obtained in [25].

    Notation 5.1. From now on, unless indicated otherwise, we denote by U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}} an open subset with the property from Lemma 3.8.

    We first determine the gradient of a smooth function on f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} with respect to the metric \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathrm{St}_{{{n}}, {{k}}}) \big) . Let \sharp_{D, E} \colon T_X^* \mathrm{St}_{{{n}}, {{k}}} \to T_X \mathrm{St}_{{{n}}, {{k}}} denote the sharp map associated with \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} , i.e. the inverse of the flat map \flat \colon T_X \mathrm{St}_{{{n}}, {{k}}} \ni V \mapsto \langle V, \cdot \rangle_X^{D, E} \in T_X^* \mathrm{St}_{{{n}}, {{k}}} . Then \mathrm{grad} f \in \Gamma^{\infty}(T \mathrm{St}_{{{n}}, {{k}}}) is the unique vector field that fulfills

    \begin{equation} \big\langle \mathrm{grad} f(X) , V \big\rangle_X^{D, E} = {\rm{d}}\ f\big\vert_{{X}}(V) \quad \iff \quad \mathrm{grad} f(X) = \big( {\rm{d}}\ f\big\vert_{{X}}(\cdot)\big)^{\sharp_{D, E}} \end{equation} (5.1)

    for all X \in \mathrm{St}_{{{n}}, {{k}}} and V \in T_X \mathrm{St}_{{{n}}, {{k}}} , see e.g. [26, Sec. 8.1] for the Riemannian case, which clearly extends to the pseudo-Riemannian case.

    Proposition 5.2. Let f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} be smooth with some smooth extension F \colon U \to \mathbb{{{R}}} . Then the gradient of f at X \in \mathrm{St}_{{{n}}, {{k}}} with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} is given by

    \begin{equation} \mathrm{grad} f(X) = \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) D^{-1} + X \pi^{D + E}\big(X^{\top} \nabla F(X) \big( D^{-1} - (D + E)^{-1} E D^{-1}\big)\big) . \end{equation} (5.2)

    Proof. We first compute the gradient of F \colon U \to \mathbb{{{R}}} with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} . Let X \in U . Then \mathrm{grad} F(X) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} fulfills

    \begin{equation} \big\langle \mathrm{grad} F(X) , V \big\rangle_X^{D, E} = {\rm{d}}\ F\big\vert_{{X}}(V) = {\rm{tr}}\big( \big( \nabla F(X) \big)^{\top} V \big) \end{equation} (5.3)

    for all V \in T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}} . Using the definition of \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} , Equation (5.3) can be rewritten as

    \begin{equation*} {\rm{tr}}\big( V^{\top} \big( \mathrm{grad} F(X) D + X X^{\top} \mathrm{grad} F(X) E\big)\big) = {\rm{tr}}\big(V^{\top} \nabla F(X) \big). \end{equation*}

    Since \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} is non-degenerated, this is equivalent to the linear equation

    \begin{equation} \mathrm{grad} F(X) D + X X^{\top} \mathrm{grad} F(X) E = \nabla F(X) \end{equation} (5.4)

    in terms of \mathrm{grad} F(X) . Now assume X \in \mathrm{St}_{{{n}}, {{k}}} . Then the unique solution of (5.4) is given by

    \begin{equation*} \mathrm{grad} F(X) = \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) (D + E)^{-1} E D^{-1} \end{equation*}

    according to Lemma 4.8. Next, we use the well-known formula \mathrm{grad} f(X) = P_X \big(\mathrm{grad} F(X) \big) , where P_X \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to T_X \mathrm{St}_{{{n}}, {{k}}} is determined in Theorem 3.18. One calculates

    \begin{equation*} \begin{split} \mathrm{grad} f(X) & = P_X\big( \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) (D + E)^{-1} E D^{-1} \big) \\ & = \Big( \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) D^{-1} + X \pi^{D + E}\big(X^{\top} \nabla F(X) D^{-1}\big) \Big) \\ &\quad - \Big( X X^{\top} \nabla F(X) (D + E)^{-1} E D^{-1} - X X^{\top} \big( X X^{\top} \nabla F(X) (D + E)^{-1} E D^{-1} \big) \\ &\qquad + X \pi^{D + E}\big( X^{\top} X X^{\top} \nabla F(X) (D + E)^{-1} E D^{-1} \big)\Big) \\ & = \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) D^{-1} + X \pi^{D + E}\big(X^{\top} \nabla F(X) \big( D^{-1} - (D + E)^{-1} E D^{-1}\big)\big) \end{split} \end{equation*}

    for X \in \mathrm{St}_{{{n}}, {{k}}} as desired.

    Next we specialize the formula for the gradient to the subfamily \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} .

    Proposition 5.3. Let f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} be smooth with some smooth extension F \colon U \to \mathbb{{{R}}} . Then the gradient of f with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} is given by

    \begin{equation} \mathrm{grad} f(X) = \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) D^{-1} + X \pi^{D + \nu I_k}\big(X^{\top} \nabla F(X) (D + \nu I_k)^{-1}\big) \end{equation} (5.5)

    for all X \in \mathrm{St}_{{{n}}, {{k}}} .

    Proof. Using Proposition 5.2 we obtain for X \in T_X \mathrm{St}_{{{n}}, {{k}}}

    \begin{equation*} \begin{split} \mathrm{grad} f(X) & = \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) D^{-1} + X \pi^{D + \nu I_k}\big(X^{\top} \nabla F(X) \big( D^{-1} - \nu (D + \nu I_k)^{-1} D^{-1} \big)\big) \\ & = \nabla F(X) D^{-1} - X X^{\top} \nabla F(X) D^{-1} + X \pi^{D + \nu I_k}\big(X^{\top} \nabla F(X) (D + \nu I_k)^{-1}\big), \end{split} \end{equation*}

    where the identity

    \begin{equation*} \big( D^{-1} - \nu (D + \nu I_k)^{-1} D^{-1} \big)_{ii} = \tfrac{1 }{D_{ii}} - \tfrac{\nu}{(D_{ii} + \nu) D_{ii}} = \tfrac{D_{ii} + \nu - \nu }{(D_{ii} + \nu)D_{ii}} = \big((D + \nu I_k)^{-1} \big)_{ii} \end{equation*}

    is used to obtain the last equality.

    Corollary 5.4. Let \alpha \in T_X^* \mathrm{St}_{{{n}}, {{k}}} be given by

    \begin{equation} \alpha = {\rm{tr}}\big( V^{\top} (\cdot) \big) \colon T_X \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}}, \quad W \mapsto {\rm{tr}}(V^{\top} W) \in \mathbb{{{R}}}, \end{equation} (5.6)

    where V \in \mathbb{{{R}}}^{{{n}} \times {{k}}} is some matrix. The sharp map \sharp_{D, E} \colon T_X^* \mathrm{St}_{{{n}}, {{k}}} \to T_X \mathrm{St}_{{{n}}, {{k}}} associated with \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} applied to \alpha is given by

    \begin{equation} \alpha^{\sharp_{D, E}} = \big({\rm{tr}}\big(V^{\top}(\cdot)\big) \big)^{\sharp_{D, \nu I_k}} = V D^{-1} - X X^{\top} V D^{-1} + X \pi^{D + E}\big(X^{\top} V \big( D^{-1} - (D + E)^{-1} E D^{-1}\big) \big). \end{equation} (5.7)

    Specializing E = \nu I_k yields the sharp map with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} applied to \alpha , namely

    \begin{equation} \alpha^{\sharp_{D, \nu I_k}} = \big({\rm{tr}}\big(V^{\top}(\cdot)\big) \big)^{\sharp_{D, \nu I_k}} = V D^{-1} - X X^{\top} V D^{-1} + X \pi^{D + \nu I_k}\big(X^{\top} V (D + \nu I_k)^{-1}\big) . \end{equation} (5.8)

    Proof. Consider the smooth function F \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \ni X \mapsto {\rm{tr}}\big(V^{\top} X \big) \in \mathbb{{{R}}} and set f = F\big\vert_{{\mathrm{St}_{{{n}}, {{k}}}}}\colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} . Then {\rm{d}}\ F\big\vert_{{X}}(W) = {\rm{tr}}(V^{\top} W) and thus \nabla F(X) = V follows. Applying Proposition 5.2 and Proposition 5.3, respectively, yields the desired result because of (5.1).

    Corollary 5.5. Proposition 5.3 reproduces some results known from the literature as special cases:

    1. For D = I_k and \nu = 0 one has

    \begin{equation} \mathrm{grad} f(X) = \nabla F(X) - \tfrac{1}{2} X X^{\top} \nabla F(X) - \tfrac{1}{2} X \big(\nabla F(X) \big)^{\top} X . \end{equation} (5.9)

    This coincides with the gradient with respect to the Euclidean metric, see e.g. [2].

    2. For D = I_k and \nu = - \tfrac{1}{2} , one has

    \begin{equation} \mathrm{grad} f(X) = \nabla f(X) - X \big(\nabla F(X) \big)^{\top} X \end{equation} (5.10)

    reproducing the formula for the gradient from [10, Eq. (2.53)].

    3. For D = 2 I_k and -2 \neq \nu \in \mathbb{{{R}}} the gradient of f simplifies to

    \begin{equation} \mathrm{grad} f(X) = \tfrac{1}{2} \big( \nabla f(X) - \tfrac{\nu + 1}{2 + \nu} X X^{\top} \nabla F(X) - \tfrac{1}{2 + \nu} X \big(\nabla F(X) \big)^{\top} X \big) \end{equation} (5.11)

    reproducing the expression for the gradient from [14, Thm. 1].

    Proof. These formulas follow by straightforward calculations by plugging the particular choices for D and \nu into the expression for \mathrm{grad} f from Proposition 5.3.

    Next we determine the pseudo-Riemannian Hessian of a smooth function f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} . Here we only consider the subfamily \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} in order to obtain formulas which are not too complicated.

    Lemma 5.6. Let X \in \mathrm{St}_{{{n}}, {{k}}} , V \in T_X \mathrm{St}_{{{n}}, {{k}}} and let f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} be smooth with some smooth extension F \colon U \to \mathbb{{{R}}} . The Hessian of f with respect \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} considered as quadratic form is given by

    \begin{equation} \begin{split} \mathrm{Hess}(f)\big\vert_{{X}}(V, V) & = {\rm{D}}^2 F(X)(V, V) \\ &\quad + {\rm{D}} F(X) \big( 2 \nu V V^{\top} X D^{-1} + 2 \nu X (X^{\top} V)^2 D^{-1} - X V^{\top} V + X \pi^{D + \nu I_k}\big(V^{\top} V \big) \big) , \end{split} \end{equation} (5.12)

    where X \in \mathrm{St}_{{{n}}, {{k}}} and V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} .

    Proof. The geodesic \gamma \colon \mathbb{{{R}}} \to \mathrm{St}_{{{n}}, {{k}}} through \gamma(0) = X \in \mathrm{St}_{{{n}}, {{k}}} with initial velocity \dot{\gamma}(0) = V \in T_X \mathrm{St}_{{{n}}, {{k}}} fulfills the explicit second order ODE

    \begin{equation} \begin{split} \ddot{\gamma}(t) & = 2 \nu \dot{\gamma}(t) \dot{\gamma}(t)^{\top} \gamma(t) D^{-1} + 2 \nu \gamma(t) \big(\gamma(t)^{\top} \dot{\gamma}(t) \big)^2 D^{-1} \\ &\quad - \gamma(t) \dot{\gamma}(t)^{\top} \dot{\gamma}(t) + \gamma(t) \pi^{D + \nu I_k}\big(\dot{\gamma}(t)^{\top} \dot{\gamma}(t) \big) \end{split} \end{equation} (5.13)

    according to Corollary 4.18. Evaluating (5.13) at t = 0 yields

    \begin{equation} \ddot{\gamma}(0) = 2 \nu V V^{\top} X D^{-1} + 2 \nu X (X^{\top} V)^2 D^{-1} - X V^{\top} V + X \pi^{D + \nu I_k}\big(V^{\top} V \big) \end{equation} (5.14)

    due to the initial conditions \gamma(0) = X and \dot{\gamma}(0) = V . The Hessian of f considered as quadratic form can be determined as

    \begin{equation} \mathrm{Hess}(f)\big\vert_{{X}}(V, V) = \tfrac{{\rm{d}}\ ^2}{{\rm{d}}\ t^2} (f \circ \gamma)(t) \big\vert_{{t = 0}}, \end{equation} (5.15)

    see e.g. [26, Prop. 8.3] for the Riemannian case, which clearly extends to pseudo-Riemannian manifolds. Using f = F\big\vert_{{\mathrm{St}_{{{n}}, {{k}}}}} , Formula (5.15) yields

    \begin{equation} \mathrm{Hess}(f)\big\vert_{{X}}(V, V) = {\rm{D}}^2 F(X)\big(\dot{\gamma}(0), \dot{\gamma}(0)\big) + {\rm{D}} F(X) \ddot{\gamma}(0) \end{equation} (5.16)

    by the chain rule. Plugging (5.14) into (5.16) yields the desired result.

    Theorem 5.7. Let X \in \mathrm{St}_{{{n}}, {{k}}} and V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} . Moreover, define \widetilde{D} = D + \nu I_k . The Hessian of a smooth function f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} with smooth extension F \colon U \to \mathbb{{{R}}} with respect \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} is given by

    \begin{equation} \begin{split} \mathrm{Hess}(f)\big\vert_{{X}}(V, W) & = {\rm{tr}}\big( \big( {\rm{D}} (\nabla F)(X) V \big)^{\top} W \big) \\ &\quad +\nu {\rm{tr}} \big( \big( X D^{-1} (\nabla F(X))^{\top} V + \nabla F(X) D^{-1} X^{\top} V \big)^{\top} W \big) \\ &\quad +\nu {\rm{tr}}\big( \big( X V^{\top} X X^{\top} \nabla F(X) D^{-1} + X X^{\top} \nabla F(X) D^{-1} V^{\top} X \big)^{\top} W \big) \\ &\quad - \tfrac{1}{2} {\rm{tr}}\big( \big( V X^{\top} \nabla F(X) + V \big( \nabla F(X) \big)^{\top} X \big)^{\top} W \big) \\ &\quad + \tfrac{1}{2} {\rm{tr}}\big( \big( V \pi^{\widetilde{D}} (X^{\top} \nabla F(X) \widetilde{D}^{-1}) \widetilde{D} - V \widetilde{D} \pi^{\widetilde{D}}(X^{\top} \nabla F(X) \widetilde{D}^{-1}) \big)^{\top} W \big) . \end{split} \end{equation} (5.17)

    Proof. Let (X, V), (X, W) \in T \mathrm{St}_{{{n}}, {{k}}} . We obtain for the Hessian of f as symmetric 2 -tensor

    \begin{equation} \begin{split} \mathrm{Hess}(f)\big\vert_{{X}}(V, W) & = {\rm{tr}}\big( \big( {\rm{D}} (\nabla F)(X) V \big)^{\top} W\big) \\ &\quad +\nu {\rm{tr}}\big( \big(\nabla F(X)\big)^{\top} \big(V W^{\top} + W V^{\top}\big) X D^{-1} \big) \\ &\quad +\nu {\rm{tr}}\big( \big(\nabla F(X)\big)^{\top} X \big( X^{\top} V X^{\top} W + X^{\top} W X^{\top} V \big) D^{-1} \big) \\ &\quad - \tfrac{1}{2} {\rm{tr}}\big( \big(\nabla F(X)\big)^{\top} X \big(V^{\top} W + W^{\top} V\big) \big) \\ &\quad + \tfrac{1}{2} {\rm{tr}}\big( \big(\nabla F(X) \big)^{\top} X \pi^{D + \nu I_k}(V^{\top} W + W^{\top} V) \big) \end{split} \end{equation} (5.18)

    by applying polarization to the quadratic form obtained in Lemma 5.6 and using the identities

    \begin{equation*} {\rm{D}} F(X) V = {\rm{tr}} \big( \big(\nabla F(X))^{\top} V \big) \quad \ {\text{and}} \quad {\rm{D}}^2 F(X)(V, W) = {\rm{tr}} \big( \big({\rm{D}} (\nabla F) (X) V \big)^{\top} W \big) . \end{equation*}

    Next, we set \widetilde{D} = D + \nu I_k which is invertible according to Notation 3.13. Since the orthogonal projection \pi^{\widetilde{D}} \colon \mathbb{{{R}}}^{{{k}} \times {{k}}} \to \mathfrak{{{so}}}(k) \subseteq \mathbb{{{R}}}^{{{k}} \times {{k}}} is self-adjoint with respect to the scalar product

    \begin{equation*} \langle \cdot, \cdot \rangle^{\widetilde{D}} \colon \mathbb{{{R}}}^{{{k}} \times {{k}}} \times \mathbb{{{R}}}^{{{k}} \times {{k}}} \to \mathbb{{{R}}}, \quad (V, W) \mapsto {\rm{tr}}\big(V^{\top} W \widetilde{D} \big) \end{equation*}

    on \mathbb{{{R}}}^{{{k}} \times {{k}}} , we calculate

    \begin{equation} \begin{split} {\rm{tr}}\big( \big(\nabla F(X) \big)^{\top} X \pi^{\widetilde{D}}\big(V^{\top} W + W^{\top} V \big) \big) & = {\rm{tr}}\big( \big( X^{\top} \nabla F(X) \widetilde{D}^{-1} \big)^{\top} \pi^{\widetilde{D}}\big(V^{\top} W + W^{\top} V \big) \widetilde{D} \big) \\ & = \big\langle X^{\top} \nabla F(X) \widetilde{D}^{-1}, \pi^{\widetilde{D}}\big(V^{\top} W + W^{\top} V \big) \big\rangle^{\widetilde{D}} \\ & = \big\langle \pi^{\widetilde{D}}\big( X^{\top} \nabla F(X) \widetilde{D}^{-1}\big), V^{\top} W + W^{\top} V \big\rangle^{\widetilde{D}} \\ & = {\rm{tr}}\big( \big( V \pi^{\widetilde{D}}\big( X^{\top} \nabla F(X) \widetilde{D}^{-1} \big) \widetilde{D} - V \widetilde{D} \pi^{\widetilde{D}}\big( X^{\top} \nabla F(X) \widetilde{D}^{-1} \big) \big)^{\top} W \big) \end{split} \end{equation} (5.19)

    by exploiting {\rm{im}}(\pi^{\widetilde{D}}) = \mathfrak{{{so}}}(k) . The desired result follows by rewriting (5.18) using well-known properties of the trace and applying (5.19) to the last summand of (5.18).

    Corollary 5.8. 5.7: Let D = 2 I_k and - 2 \neq \nu \in \mathbb{{{R}}} . Then the Hessian of the smooth function f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} reads

    \begin{equation} \begin{split} \mathrm{Hess}(f)\big\vert_{{X}}(V, W) & = {\rm{tr}}\big( \big( {\rm{D}} (\nabla F)(X) V \big)^{\top} W \big) \\ &\quad +\tfrac{\nu}{2} {\rm{tr}}\big( \big( X \big(\nabla F(X) \big)^{\top} V + \nabla F(X) X^{\top} V \big)^{\top} W \big) \\ &\quad +\tfrac{\nu}{2} {\rm{tr}}\big( \big( X V^{\top} X X^{\top} \nabla F(X) + X X^{\top} \nabla F(X) V^{\top} X \big)^{\top} W \big) \\ &\quad - \tfrac{1}{2} {\rm{tr}}\big( \big( V X^{\top} \nabla F(X) + V \big( \nabla F(X) \big)^{\top} X \big)^{\top} W \big) \end{split} \end{equation} (5.20)

    with X \in \mathrm{St}_{{{n}}, {{k}}} and V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} reproducing the formula from [14, Thm. 2].

    Proof. We set D = 2 I_k in Theorem 5.7. Obviously, \widetilde{D} = (2 + \nu) I_k holds. Hence

    \begin{equation*} \pi^{\widetilde{D}} \big(X^{\top} \nabla F(X) \widetilde{D}^{-1} \big) \widetilde{D} = \pi^{\widetilde{D}} \big(X^{\top} \nabla F(X) \big) = \widetilde{D} \pi^{\widetilde{D}} \big(X^{\top} \nabla F(X) \widetilde{D}^{-1} \big) \end{equation*}

    is fulfilled by the linearity of \pi^{\widetilde{D}} \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathfrak{{{so}}}(k) \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}} . Thus the last summand of (5.17) vanishes.

    Theorem 5.7 yields an expression for the Hessian of f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} as covariant 2 -tensor. However, for applications, see e.g. [2, Chap. 6], an expression for the Hessian of f viewed as section of \mathrm{End}(T \mathrm{St}_{{{n}}, {{k}}}) is desirable. Thus, following [14, Re. 6], we state the next remark and the next corollary.

    Remark 5.9. In [2, Eq. (6.3)] the Hessian of a smooth function f \colon M \to \mathbb{{{R}}} on a Riemannian manifold \big(M, \langle \cdot, \cdot \rangle\big) endowed with a covariant derivative \nabla is defined as

    \begin{equation*} \widetilde{ \mathrm{Hess}}(f)\big\vert_{{x}}(v_x) = \nabla_{v_x} \mathrm{grad} f\big\vert_{{x}} \end{equation*}

    for x \in M and v_x \in T_x M . In particular, \widetilde{ \mathrm{Hess}}(f) \in \Gamma^{\infty}\big(\mathrm{End}(T M) \big) holds. If \nabla is chosen as the Levi-Civita covariant derivative \nabla^{\mathrm{LC}} , then \widetilde{ \mathrm{Hess}}(f) is related to \mathrm{Hess}(f) \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* M)\big) via

    \begin{equation} \big\langle \widetilde{ \mathrm{Hess}}(f)\big\vert_{{x}}(v), w \big\rangle = \big\langle \nabla^{\mathrm{LC}}_{v_x} \mathrm{grad} f \big\vert_{{x}}, w_x \big\rangle = \mathrm{Hess}(f)\big\vert_{{x}}(v_x, w_x), \end{equation} (5.21)

    where x \in M and v_x, w_x \in T_x M , see e.g. [26, Prop. 8.1] for a proof for the Riemannian case. Clearly, Equation (5.21) holds in the pseudo-Riemannian case, too. We rewrite (5.21) equivalently as

    \begin{equation} \big\langle \widetilde{ \mathrm{Hess}}(f)\big\vert_{{x}}(v_x), \cdot \big\rangle = \mathrm{Hess}(f)\big\vert_{{x}}(v_x, \cdot) . \end{equation} (5.22)

    Applying the sharp map \sharp \colon T_x^* M \to T_x M associated with \langle \cdot, \cdot \rangle on both sides of (5.22) yields

    \begin{equation} \widetilde{ \mathrm{Hess}}(f)\big\vert_{{x}}(v_x) = \big( \mathrm{Hess}(f)\big\vert_{{x}}(v_x, \cdot)\big)^{\sharp} . \end{equation} (5.23)

    Corollary 5.10. Let f \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}} be smooth with some smooth extension F \colon U \to \mathbb{{{R}}} . The Hessian of f with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} considered as a section of \mathrm{End}(T \mathrm{St}_{{{n}}, {{k}}}) is given by

    \begin{equation*} \begin{split} \widetilde{ \mathrm{Hess}}(f)\big\vert_{{X}}(V) & = L_X^{D, \nu} \Big( {\rm{D}} (\nabla F)(X) V +\nu \big( X D^{-1} (\nabla F(X))^{\top} V + \nabla F(X) D^{-1} X^{\top} V \big) \\ &\quad +\nu \big( X V^{\top} X X^{\top} \nabla F(X) D^{-1} + X X^{\top} \nabla F(X) D^{-1} V^{\top} X \big) \\ &\quad - \tfrac{1}{2} \big( V X^{\top} \nabla F(X) + V \big( \nabla F(X) \big)^{\top} X \big) \\ &\quad + \tfrac{1}{2} \big( V \pi^{\widetilde{D}}\big(X^{\top} \nabla F(X) \widetilde{D}^{-1} \big) \widetilde{D} - V \widetilde{D}\pi^{\widetilde{D}} \big(X^{\top} \nabla F(X) \widetilde{D}^{-1} \big) \big) \Big) \end{split} \end{equation*}

    for X \in \mathrm{St}_{{{n}}, {{k}}} and V \in T_X \mathrm{St}_{{{n}}, {{k}}} , where \widetilde{D} = D + \nu I_k and L_X^{D, \nu} \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to T_X \mathrm{St}_{{{n}}, {{k}}} \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}} is the linear map given by

    \begin{equation*} L_X^{D, \nu}(V) = V D^{-1} - X X^{\top} V D^{-1} + X \pi^{D + \nu I_k}\big(X^{\top} V (D + \nu I_k)^{-1}\big) . \end{equation*}

    Proof. We have already obtained \mathrm{Hess}(f) in Theorem 5.7 in such a form that the formula for the sharp map from Corollary 5.4 can be applied to \mathrm{Hess}(f)\big\vert_{{X}}(V, \cdot) \in T_X^*\mathrm{St}_{{{n}}, {{k}}} . Now Remark 5.9 yields the desired result.

    In this section, we compute the second fundamental form of \mathrm{St}_{{{n}}, {{k}}} considered as pseudo-Riemannian submanifold of \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big) . Moreover, an expression for the Levi-Civita covariant derivative on \mathrm{St}_{{{n}}, {{k}}} is derived. We first recall Notation 5.1. Unless indicated otherwise, we denote by U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}} an open neighbourhood of \mathrm{St}_{{{n}}, {{k}}} with the property from Lemma 3.8.

    We consider the Levi-Civita covariant derivative \widetilde{ \nabla^{\mathrm{LC}}} on U with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} . Recall Proposition 4.9. For (X, V) \in TU the spray \widetilde{S} \in \Gamma^{\infty}\big(T (TU)\big) associated with the metric \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} on U is given by

    \begin{equation} \widetilde{S}(X, V) = \big(X, V, V, - \widetilde{\Gamma}_X(V, V) \big) \end{equation} (6.1)

    where \widetilde{\Gamma}_X(V, V) is the unique solution of the linear equation (4.11). We now discuss how \widetilde{\Gamma}_X(V, V) \! \in \! \mathbb{{{R}}}^{{{n}} \times {{k}}} is related to the Christoffel symbols of the Levi-Civita covariant derivative \widetilde{ \nabla^{\mathrm{LC}}} on \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) . To this end, we view \mathrm{id}_U \colon U \ni X \mapsto X \in U as the global chart \big(U, \mathrm{id}_U \big) = \big(U, X_{ij} \big) of U and identify the coordinate vector fields \frac{\partial}{\partial X_{ij}} with the constant functions U \ni X \mapsto E_{ij} \in \mathbb{{{R}}}^{{{n}} \times {{k}}} . Then (6.1) is a coordinate expression for the metric spray \widetilde{S} with respect to the global chart \big(TU, (X_{ij}, V_{ij}) \big) induced by the chart (U, X_{ij}) , see also Proposition 4.9. Thus the local form of metric sprays, see (4.2), implies that the entry \big(\widetilde{\Gamma}_X(V, V)\big)_{ij} fulfills

    \begin{equation} \big(\widetilde{\Gamma}_X(V, V)\big)_{ij} = \sum\limits_{a, c = 1}^n \sum\limits_{b, d = 1}^k \widetilde{\Gamma}_{(a, b), (c, d)}^{(i, j)} \big\vert_{{X}}V_{a b} V_{c d}, \end{equation} (6.2)

    where V = (V_{ij}) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} and the functions \widetilde{\Gamma}_{(a, b), (c, d)}^{(i, j)} \colon U \ni X \mapsto \widetilde{\Gamma}_{(a, b), (c, d)}^{(i, j)}\big\vert_{{X}}\in \mathbb{{{R}}} denote the Christoffel symbols of \widetilde{ \nabla^{\mathrm{LC}}} with respect to \big(U, (X_{ij}) \big) . Hence \widetilde{ \nabla^{\mathrm{LC}}} can be expressed with respect to the global chart (U, X_{ij}) as

    \begin{equation} \widetilde{ \nabla^{\mathrm{LC}}_{\widetilde{V}}} \widetilde{W} \big\vert_{{X}} = {\rm{D}} \widetilde{W}(X) \widetilde{V}\big\vert_{{X}}+ \widetilde{\Gamma}_X\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big), \end{equation} (6.3)

    for vector fields \widetilde{V}, \widetilde{W} \in \Gamma^{\infty}(T U) and X \in U , see e.g. [27, Chap. 4]. A similar "matrix notation" for Christoffel symbols has already appeared in [10, Sec. 2.2.3], where, in addition, it is mentioned that (for fixed X \in U ) the symmetric bilinear map \mathbb{{{R}}}^{{{n}} \times {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}} \ni (V, W) \mapsto \widetilde{\Gamma}_X(V, W) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} can be obtained from the quadratic map \mathbb{{{R}}}^{{{n}} \times {{k}}} \ni V \mapsto \widetilde{\Gamma}_X(V, V) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} by polarization. Hence the Christoffel symbols on U can be identified with the smooth map

    \begin{equation} \widetilde{\Gamma} \colon U \to \mathrm{S}^2\big((\mathbb{{{R}}}^{{{n}} \times {{k}}})^* \big) \otimes \mathbb{{{R}}}^{{{n}} \times {{k}}} , \quad X \mapsto \big((V, W) \mapsto \widetilde{\Gamma}_X(V, W) \big) . \end{equation} (6.4)

    The "Christoffel symbols" from [10, Sec. 2.2.3] will be discussed in Remark 6.11 below.

    Next we give an expression for the Levi-Civita covariant derivative \nabla^{\mathrm{LC}} on \mathrm{St}_{{{n}}, {{k}}} with respect \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} . We refer to Proposition 6.8 as well as Corollary 6.9 below for an alternative formula for \nabla^{\mathrm{LC}} .

    Proposition 6.1. Let V, W \in \Gamma^{\infty}(T \mathrm{St}_{{{n}}, {{k}}}) . The Levi-Civita covariant derivative on \big(\mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) is given by

    \begin{equation} \nabla^{\mathrm{LC}}_V W \big\vert_{{X}} = P_X\Big( {\rm{D}} \widetilde{W}(X) V \big\vert_{{X}}+ \widetilde{\Gamma}_X\big( V\big\vert_{{X}}, W\big\vert_{{X}}\big) \Big) \end{equation} (6.5)

    for all X \in \mathrm{St}_{{{n}}, {{k}}} , where \widetilde{V} \in \Gamma^{\infty}(T U) is a smooth extensions of V . Here \widetilde{\Gamma} is defined by (6.4). Moreover, P_X \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to T_X \mathrm{St}_{{{n}}, {{k}}} is the orthogonal projection with respect to \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} from Theorem 3.18.

    Proof. Since \mathrm{St}_{{{n}}, {{k}}} is a pseudo-Riemannian submanifold of \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) , the result follows by (6.3) due to

    \begin{equation*} \nabla^{\mathrm{LC}}_V W \big\vert_{{X}} = P_X \Big(\widetilde{ \nabla^{\mathrm{LC}}_{V}} \widetilde{W}\big\vert_{{X}}\Big) , \end{equation*}

    see e.g. [17, Chap. 4, Lem. 3].

    We now consider the second fundamental form, also called shape operator, of \mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) . We refer to [17, Chap. 4] for general properties of pseudo-Riemanian submanifolds and the second fundamental form, see also [27, Chap. 8] for the Riemannian case. Using these references, we briefly introduce the notation which is used in the sequel subsections.

    Let M be a pseudo-Riemannian submanifold of a pseudo-Riemannian manifold \big(\widetilde{M}, \langle \cdot, \cdot \rangle \big) . The corresponding Levi-Civita covariant derivatives on M and \widetilde{M} are denoted by \nabla^{\mathrm{LC}} and \widetilde{ \nabla^{\mathrm{LC}}} , respectively. Moreover, let N M \to M be the normal bundle of M and let \mathrm{II} \in \Gamma^{\infty}\big((\mathrm{S}^2 (T^* M)) \otimes N M \big) be the second fundamental form of M , see e.g. [17, Chap. 4, Lem. 4], defined by

    \begin{equation} \mathrm{II}(V, W)\big\vert_{{x}} = P_x^{\perp}\Big(\widetilde{ \nabla^{\mathrm{LC}}_{\widetilde{V}}} \widetilde{W}\big\vert_{{x}}\Big), \quad x \in M, \quad V, W \in \Gamma^{\infty}(T M) , \end{equation} (6.6)

    where \widetilde{V}, \widetilde{W} \in \Gamma^{\infty}(T \widetilde{M}) are smooth extensions of V, W \in \Gamma^{\infty}(T M) , respectively, and P_x^{\perp} \colon T_x \widetilde{M} \to N_x M denotes the orthogonal projection onto the normal space N_x M = (T_x M)^{\perp} . The Levi-Civita covariant derivative on M fulfills

    \begin{equation} \nabla^{\mathrm{LC}}_V W = \widetilde{ \nabla^{\mathrm{LC}}_V} \widetilde{W} - \mathrm{II}(V, W) \end{equation} (6.7)

    for all V, W \in \Gamma^{\infty}(TM) by [17, Chap. 4]. Here \widetilde{W} is again some smooth extension of W to \widetilde{M} . The identity (6.7) is named Gauß formula in [27, Thm. 8.2], which includes a proof for the Riemannian case, as well.

    Lemma 6.2. Define \nabla \colon \Gamma^{\infty}(T \widetilde{M}) \times \Gamma^{\infty}(T \widetilde{M}) \to \Gamma^{\infty}(T \widetilde{M}) by

    \begin{equation} \nabla_{\widetilde{V}} \widetilde{W} = \widetilde{ \nabla^{\mathrm{LC}}_{\widetilde{V}}} \widetilde{W} - \widetilde{ \mathrm{II}}(\widetilde{V}, \widetilde{W}), \quad \widetilde{V}, \widetilde{W} \in \Gamma^{\infty}(T \widetilde{M}), \end{equation} (6.8)

    where \widetilde{ \mathrm{II}} \in \Gamma^{\infty}\big((\mathrm{S}^2 T^* \widetilde{M}) \otimes T \widetilde{M} \big) denotes a smooth extension of the second fundamental form \mathrm{II} on M to \widetilde{M} . Then \nabla is a covariant derivative on \widetilde{M} whose restriction to M coincides with \nabla^{\mathrm{LC}} , i.e.

    \begin{equation} \nabla^{\mathrm{LC}}_V W \big\vert_{{x}} = \nabla_{\widetilde{V}} \widetilde{W} \big\vert_{{x}} \end{equation} (6.9)

    holds for all x \in M and V, W \in \Gamma^{\infty}(TM) with smooth extensions \widetilde{V}, \widetilde{W} \in \Gamma^{\infty}(T \widetilde{M}) . Moreover, the Christoffel symbols of \nabla with respect to the local chart (U, x) of \widetilde{M} are given by

    \begin{equation} \Gamma_{ij}^k = \widetilde{\Gamma}_{ij}^k - \widetilde{ \mathrm{II}}_{ij}^k . \end{equation} (6.10)

    Here \widetilde{ \mathrm{II}}_{ij}^k is defined by \widetilde{ \mathrm{II}}\big(\tfrac{\partial}{\partial x^i}, \tfrac{\partial}{\partial x^j} \big) = \widetilde{ \mathrm{II}}_{ij}^k \tfrac{\partial}{\partial x^k} using Einstein summation convention and \widetilde{\Gamma}_{ij}^k denote the Christoffel symbols of \widetilde{ \nabla^{\mathrm{LC}}} with respect to the chart (U, x) .

    Proof. Obviously, the definition of \nabla yields a covariant derivative on M . Moreover, the Gauß formula (6.7) implies

    \begin{equation*} ´ \nabla^{\mathrm{LC}}_{\widetilde{V}} \widetilde{W}\big\vert_{{x}} = \widetilde{ \nabla^{\mathrm{LC}}_{\widetilde{V}}} \widetilde{W} \big\vert_{{x}}- \mathrm{II}\big\vert_{{x}}\big(\widetilde{V}\big\vert_{{x}}, \widetilde{W}\big\vert_{{x}}\big) = \nabla_{\widetilde{V}} \widetilde{W}\big\vert_{{x}} \end{equation*}

    for all x \in M and all V, W \in \Gamma^{\infty}(T M) with smooth extensions \widetilde{V}, \widetilde{W} \in \Gamma^{\infty}(T \widetilde{M}) , respectively.

    It remains to show the formula for the Christoffel symbols. Let (U, x) be a local chart of \widetilde{M} . Using Einstein summation convention one obtains

    \begin{equation*} \nabla_{\tfrac{\partial}{\partial x^i}} \tfrac{\partial}{\partial x^j} = \widetilde{ \nabla^{\mathrm{LC}}_{\tfrac{\partial}{\partial x^i}}} \tfrac{\partial}{\partial x^j} - \widetilde{ \mathrm{II}}\big( \tfrac{\partial}{\partial x^i}, \tfrac{\partial}{\partial x^j} \big) = \widetilde{\Gamma}_{ij}^k \tfrac{\partial}{\partial x^k} - \widetilde{ \mathrm{II}}_{ij}^k \tfrac{\partial}{\partial x^k} = \big( \widetilde{\Gamma}_{ij}^k - \widetilde{ \mathrm{II}}_{ij}^k \big) \tfrac{\partial}{\partial x^k} \end{equation*}

    showing the desired result.

    Remark 6.3. The definition of the covariant derivative \nabla on \widetilde{M} in Lemma 6.2 depends on the choice of the smooth extension \widetilde{ \mathrm{II}} of \mathrm{II} . Nevertheless, Equation (6.9) is independent of the extension \widetilde{ \mathrm{II}} of \mathrm{II} .

    Reformulating [16, Cor. 8.4.2] yields the next lemma which allows for computing the second fundamental form of \mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big) .

    Lemma 6.4. Let M \subseteq \widetilde{M} be a pseudo-Riemannian submanifold of \big(\widetilde{M}, \langle \cdot, \cdot \rangle \big) . Moreover, we denote by \widetilde{S} \in \Gamma^{\infty}\big(T (T \widetilde{M})\big) and S \in \Gamma^{\infty}\big(T (TM) \big) the metric sprays on TM and T \widetilde{M} , respectively. Then

    \begin{equation} \big(S - \widetilde{S} \big)(v_x) = \big( \mathrm{II}\big\vert_{{x}}(v_x, v_x)\big)^ \mathrm{ver}\big\vert_{{v_x}}, \end{equation} (6.11)

    holds for all x \in M and v_x \in T_x M , where

    \begin{equation*} (\cdot)^ \mathrm{ver}\big\vert_{{v_x}}\colon T_x M \to \mathrm{Ver}_{v_x}(TM) \subseteq T_{v_x}(T M) \end{equation*}

    is the vertical lift and \mathrm{II} \in \Gamma^{\infty}\big((\mathrm{S}^2 (T^* M)) \otimes N M \big) is the second fundamental form of M \subseteq \widetilde{M} .

    Proof. This is a direct consequence of [16, Cor. 8.4.2] as well as the definition \mathrm{II} recalled in (6.6).

    Lemma 6.4 applied to \mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) yields an expression for the second fundamental form.

    Proposition 6.5. Consider \mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) as pseudo-Riemannian submanifold. Then the following assertions are fulfilled:

    1. The second fundamental form of \mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big) is given by

    \begin{equation} \mathrm{II}\big\vert_{{X}}(V, W) = \widetilde{\Gamma}_X(V, W) - \Gamma_X(V, W) \end{equation} (6.12)

    for all X \in \mathrm{St}_{{{n}}, {{k}}} and V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} , where \Gamma_X and \widetilde{\Gamma}_X denote the symmetric bilinear maps associated with the quadratic maps defined by the sprays S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}}) \big) and \widetilde{S} \in \Gamma^{\infty}\big(T (TU) \big) , respectively.

    2. A smooth extension \widetilde{ \mathrm{II}} \in \Gamma^{\infty}\big((\mathrm{S}^2 (T^*U)) \otimes T U \big) of \mathrm{II} is given

    \begin{equation} \widetilde{ \mathrm{II}}\big\vert_{{X}}(V, W) = \tilde{\Gamma}_X(V, W) - \Gamma_X(V, W), \end{equation} (6.13)

    for all X \in U and V, W \in T_X U \cong \mathbb{{{R}}}^{{{n}} \times {{k}}} , Here we view \Gamma_X(V, W) as in Remark 4.16, i.e. as the smooth map \Gamma \colon U \to \mathrm{S}^2\big((\mathbb{{{R}}}^{{{n}} \times {{k}}})^*\big) \otimes \mathbb{{{R}}}^{{{n}} \times {{k}}} defined in (4.25).

    Proof. Lemma 6.4 applied to \mathrm{St}_{{{n}}, {{k}}} \subseteq (U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}) implies that

    \begin{equation} S(X, V) - \widetilde{S}(X, V) = \big( \mathrm{II}\big\vert_{{X}}(V, V) \big)^{ \mathrm{ver}}\big\vert_{{(X, V)}} \end{equation} (6.14)

    holds for all (X, V) \in T \mathrm{St}_{{{n}}, {{k}}} . The vertical lift for fixed (X, V) \in TU is the linear isomorphism

    \begin{equation*} (\cdot)^{ \mathrm{ver}}\big\vert_{{(X, V)}}\colon T U \to \mathrm{Ver}(T U)_{(X, V)} , \quad (X, W) \mapsto (X, W)^{ \mathrm{ver}}\big\vert_{{(X, V)}} = (X, V, 0, W) , \end{equation*}

    according to its local expression, see e.g. [20, Sec. 8.12]. Thus

    \begin{equation*} \mathrm{II}\big\vert_{{X}}(V, V) = -\Gamma_X(V, V) - \big( - \widetilde{\Gamma}_X(V, V) \big) = \widetilde{\Gamma}_X(V, V) -\Gamma_X(V, V) \end{equation*}

    follows by (6.14). Since the quadratic map T_X \mathrm{St}_{{{n}}, {{k}}} \ni V \mapsto \widetilde{\Gamma}_X(V, V) - \Gamma_X(V, V) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} determines uniquely the associated symmetric billinear map, Claim 1 is shown. Now Claim 2 is obvious.

    The second fundamental from can be simplified for all metrics in the subfamily \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} .

    Corollary 6.6. The second fundamental form of \mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} \big) is given by

    \begin{equation} \begin{split} \mathrm{II}\big\vert_{{X}}(V, W) & = - \tfrac{1}{2}X\big( V^{\top} W + W^{\top} V \big) D \big(D + \nu I_k \big)^{-1} + \nu X \big(X^{\top} V X^{\top} W + X^{\top} W X^{\top} V\big) \big(D + \nu I_k \big)^{-1} \\ &\quad + \tfrac{1}{2} X \pi^{D + \nu I_k}\big(V^{\top} W + W^{\top} V\big) \\ \end{split} \end{equation} (6.15)

    for all X \in \mathrm{St}_{{{n}}, {{k}}} and V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} .

    Proof. Let X \in \mathrm{St}_{{{n}}, {{k}}} and V \in T_X \mathrm{St}_{{{n}}, {{k}}} . We first compute the quadratic map associated with \mathrm{II} . Using Corollary 4.12 and Corollary 4.18, Proposition 6.5 implies

    \begin{equation} \begin{split} \mathrm{II}\big\vert_{{X}}(V, V) & = \widetilde{\Gamma}_X(V, V) - \Gamma_X(V, V) \\ & = \Big( 2 \nu V X^{\top} V D^{-1} + \nu X V^{\top} V \big(D + \nu I_k \big)^{-1} - 2 \nu^2 X (X^{\top} V)^2 (D + \nu I_k)^{-1} D^{-1} \Big) \\ &\quad + \Big( 2 \nu V V^{\top} X D^{-1} + 2 \nu X (X^{\top} V)^2 D^{-1} - X V^{\top} V + X \pi^{D + \nu I_k}(V^{\top} V) \Big) \\ & = X V^{\top} V \Big( \nu \big(D + \nu I_k \big)^{-1} - I_k \Big) + 2 \nu X (X^{\top} V)^2 \Big( D^{-1} - \nu(D + \nu I_k)^{-1} D^{-1} \Big) + X \pi^{D + \nu I_k}(V^{\top} V) \\ & = - X V^{\top} V D \big(D + \nu I_k \big)^{-1} + 2 \nu X (X^{\top} V)^2 \big(D + \nu I_k \big)^{-1} + X \pi^{D + \nu I_k}(V^{\top} V) . \end{split} \end{equation} (6.16)

    Here we exploited

    \begin{equation*} \big( \nu (D + \nu I_k)^{-1} - I_k \big)_{ii} = \tfrac{\nu}{D_{ii} + \nu} - 1 = \tfrac{\nu - (D_{ii} + \nu)}{D_{ii} + \nu} = - \tfrac{D_{ii}}{D_{ii} + \nu} = - \big( D (D + \nu)^{-1} \big)_{ii} \end{equation*}

    as well as

    \begin{equation*} \big( D^{-1} - \nu(D + \nu I_k)^{-1} D^{-1} \big)_{ii} = \tfrac{1}{D_{ii}} \big( 1 - \tfrac{\nu}{D_{ii} + \nu} \big) = \tfrac{1}{D_{ii}} \tfrac{D_{ii} + \nu - \nu}{D_{ii} + \nu} = \big((D + \nu I_k)^{-1}\big)_{ii} . \end{equation*}

    The desired result follows by polarization.

    Corollary 6.7. For D = 2 I_k and E = \nu I_k the second fundamental form is given by

    \begin{equation} \mathrm{II}\big\vert_{{X}}(V, W) = - \tfrac{1}{2 + \nu}X\big( V^{\top} W + W^{\top} V \big) + \tfrac{\nu}{2 + \nu} X \big(X^{\top} V X^{\top} W + X^{\top} W X^{\top} V\big) \end{equation} (6.17)

    for X \in \mathrm{St}_{{{n}}, {{k}}} and V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} .

    Proof. Plugging D = 2 I_k into the formula from Corollary 6.6 the claim follows by a straightforward calculation by exploiting \pi^{2 I_k + \nu I_k} = {\rm{skew}} \colon \mathbb{{{R}}}^{{{k}} \times {{k}}} \to \mathfrak{{{so}}}(k) .

    Next we derive an alternative expression for the Levi-Civita covariant derivative on \big(\mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big) .

    Proposition 6.8. The covariant derivative \nabla on U from Lemma 6.2 fulfills for \widetilde{V}, \widetilde{W} \in \Gamma^{\infty}(T U) and X \in U

    \begin{equation} \nabla_{\widetilde{V}} \widetilde{W} \big\vert_{{X}} = \widetilde{ \nabla^{\mathrm{LC}}_{\widetilde{V}}} \widetilde{W}\big\vert_{{X}}- \widetilde{ \mathrm{II}}\big\vert_{{X}}\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big) = {\rm{D}} \widetilde{W}(X) \widetilde{V}\big\vert_{{X}}+ \Gamma_X\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big) , \end{equation} (6.18)

    where \Gamma denotes the smooth map U \to \mathrm{S}^2\big((\mathbb{{{R}}}^{{{n}} \times {{k}}})^*\big) \otimes \mathbb{{{R}}}^{{{n}} \times {{k}}} defined in (4.25). If \widetilde{V}, \widetilde{W} \in \Gamma^{\infty}(TU) are smooth extensions of V, W \in \Gamma^{\infty}(T \mathrm{St}_{{{n}}, {{k}}}) , respectively, then

    \begin{equation} \nabla^{\mathrm{LC}}_V W \big\vert_{{X}} = {\rm{D}} \widetilde{W}(X) V\big\vert_{{X}}+ \Gamma_X\big(V\big\vert_{{X}}, W\big\vert_{{X}}\big) \end{equation} (6.19)

    is satisfied for all X \in \mathrm{St}_{{{n}}, {{k}}} .

    Proof. Using Lemma 6.2 and Proposition 6.5 we compute

    \begin{equation*} \begin{split} \nabla_{\widetilde{V}} \widetilde{W}\big\vert_{{X}}& = \widetilde{ \nabla^{\mathrm{LC}}_{\widetilde{V}}} \widetilde{W} \big\vert_{{X}}- \widetilde{ \mathrm{II}}\big\vert_{{X}}\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big) \\ & = \Big( {\rm{D}} \widetilde{W}(X) \widetilde{V}\big\vert_{{X}}+ \widetilde{\Gamma}_X\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big) \Big) - \Big( \widetilde{\Gamma}_X\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big) -\Gamma_X\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big) \Big) \\ & = {\rm{D}} \widetilde{W}(X) \widetilde{V}\big\vert_{{X}}+ \Gamma_X\big(\widetilde{V}\big\vert_{{X}}, \widetilde{W}\big\vert_{{X}}\big) \end{split} \end{equation*}

    for V, W \in \Gamma^{\infty}(TU) and X \in U showing (6.18). If \widetilde{V}, \widetilde{W} are smooth extensions of V, W \in \Gamma^{\infty}(T \mathrm{St}_{{{n}}, {{k}}}) , respectively, we obtain

    \begin{equation*} \nabla_{\widetilde{V}} \widetilde{W} \big\vert_{{X}} = \nabla^{\mathrm{LC}}_V W \big\vert_{{X}} = {\rm{D}} \widetilde{W}(X) V\big\vert_{{X}}+ \Gamma_X\big(V\big\vert_{{X}}, W\big\vert_{{X}}\big) \end{equation*}

    for all X \in \mathrm{St}_{{{n}}, {{k}}} by Lemma 6.2 proving (6.19).

    Proposition 6.8 yields a more explicit formula for the subfamily \langle \cdot, \cdot \rangle_{(\cdot)}^{\nu} .

    Corollary 6.9. Let V, W \in \Gamma^{\infty}(T \mathrm{St}_{{{n}}, {{k}}}) and let \widetilde{W} \in \Gamma^{\infty}(T U) be smooth extension of W . The Levi-Civita covariant derivative on \mathrm{St}_{{{n}}, {{k}}} with respect to the metric \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu} is given by

    \begin{equation} \nabla^{\mathrm{LC}}_V W \big\vert_{{X}} = {\rm{D}} \widetilde{W}(X) V\big\vert_{{X}}+ \Gamma_X\big(V\big\vert_{{X}}, W\big\vert_{{X}}\big) \end{equation} (6.20)

    for X \in \mathrm{St}_{{{n}}, {{k}}} , where

    \begin{equation} \begin{split} \Gamma_X(V, W) & = - \nu \big(V W^{\top} + W V^{\top} \big) X D^{-1} - \nu X \big(X^{\top} V X^{\top} W + X^{\top} W X^{\top} V\big) D^{-1} \\ &\quad + \tfrac{1}{2} X \big( V^{\top} W + W^{\top} V \big) -\tfrac{1}{2} X \pi^{D + \nu I_k}\big(V^{\top} W + W^{\top} V \big) \end{split} \end{equation} (6.21)

    writing V = V\big\vert_{{X}} and W = W\big\vert_{{X}} for short.

    Proof. The quadratic map \Gamma_X \colon T_X \mathrm{St}_{{{n}}, {{k}}} \ni V \mapsto \Gamma_X(V, V) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} is determined in Corollary 4.18. The associated symmetric bilinear map T_X \mathrm{St}_{{{n}}, {{k}}} \times T_X \mathrm{St}_{{{n}}, {{k}}} \ni (V, W) \mapsto \Gamma_X(V, W) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} can be obtained by polarization. Now Proposition 6.8 yields the desired result.

    Corollary 6.9 yields an expression for the covariant derivative with respect to the family of metrics introduced in [13].

    Corollary 6.10. Using the notation from Corollary 6.9 one obtains for \nabla^{\mathrm{LC}} on \big(\mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, \nu}\big) with D = 2 I_k and -2 \neq \nu \in \mathbb{{{R}}}

    \begin{equation} \nabla^{\mathrm{LC}}_V W \big\vert_{{X}} = {\rm{D}} \widetilde{W}(X) V - \tfrac{\nu}{2} \big(V W^{\top} + W V^{\top} \big) X - \tfrac{\nu}{2} X \big(X^{\top} V X^{\top} W + X^{\top} W X^{\top} V\big) + \tfrac{1}{2} X \big( V^{\top} W + W^{\top} V \big) . \end{equation} (6.22)

    Proof. Plugging D = 2 I_k into the formula from Corollary 6.9 yields the desired result by exploiting \pi^{2 I_k + \nu I_k}(V^{\top} W + W^{\top} V) = {\rm{skew}}(V^{\top} W + W^{\top} V) = 0 for all V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} .

    By setting D = \alpha_0 I_k and E = (\alpha_1 - \alpha_0) I_k for \alpha_0, \alpha_1 \in \mathbb{{{R}}} , Corollary 6.9 reproduces [25, Eq. (5.4)], where this expression has been obtained independently. Formulas for \nabla^{\mathrm{LC}} of a similar form as in Proposition 6.8 or Corollary 6.9 have already appeared in the literature in [10,25], see also [28, Sec. 4]. In the next remark we relate the summand \Gamma_X(V, W) in these formulas to the Christoffel symbols of the covariant derivative \nabla on the open U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}} .

    Remark 6.11. Consider the smooth map \Gamma \colon U \ni X \mapsto \big((V, W) \mapsto \Gamma_X(V, W) \big) \in \mathrm{S}^2\big((\mathbb{{{R}}}^{{{n}} \times {{k}}})^*\big) \otimes \mathbb{{{R}}}^{{{n}} \times {{k}}} from (4.25) in Remark 4.16. The Christoffel symbols of the covariant derivative \! \nabla_{V} W \! = \! \widetilde{ \nabla^{\mathrm{LC}}_V} W \! \! - \! \widetilde{ \mathrm{II}}(V, W) on U with respect to (U, \mathrm{id}_U) = (U, X_{ij}) corresponds to the entries of the matrix \Gamma_X(V, W) by Proposition 6.8. More precisely, we again identify the coordinate vector field \tfrac{\partial}{\partial X_{ij}} with the map U \ni X \mapsto E_{ij} \in \mathbb{{{R}}}^{{{n}} \times {{k}}} . Then the (i, j) -entry of \Gamma_X(V, W) \in \mathbb{{{R}}}^{{{n}} \times {{k}}} is given by a formula similar to (6.2), namely

    \begin{equation} \big(\Gamma_X(V, W)\big)_{ij} = \sum\limits_{a, c = 1}^n \sum\limits_{b, d = 1}^k \Gamma_{(a, b), (c, d)}^{(i, j)} V_{ab} W_{c d} , \end{equation} (6.23)

    where \Gamma_{(a, b), (c, d)}^{(i, j)} \colon U \to \mathbb{{{R}}} are the Christoffel symbols of \nabla with respect to the global chart (U, X_{ij}) , see Lemma 6.2 and Proposition 6.8. We point out that the map \Gamma in (4.25) corresponds to the Christoffel symbols of the covariant derivative \nabla on U but it cannot correspond to the Christoffel symbols of \nabla^{\mathrm{LC}} on \mathrm{St}_{{{n}}, {{k}}} due to \dim(\mathrm{St}_{{{n}}, {{k}}}) < n k = \dim(U) . Nevertheless, if \nabla is applied to vector fields which are tangent to \mathrm{St}_{{{n}}, {{k}}} evaluated at points X \in \mathrm{St}_{{{n}}, {{k}}} , it yields the same result as \nabla^{\mathrm{LC}} by Proposition 6.8.

    A similar expression for "Christoffel symbols" has already appeared in [10] for the so-called canonical metric as well as for the Euclidean metric, however, without relating them to the Christoffel symbols of the covariant derivative \nabla on U . Indeed, by exploiting Corollary 6.9, for D = I_k and \nu = 0 we obtain

    \begin{equation*} \Gamma_X(V, W) = \tfrac{1}{2} X (V^{\top} W + W^{\top} V) \end{equation*}

    reproducing \Gamma_e in [10, Sec. 2.2.3]. Analogously, setting D = I_k and \nu = - \tfrac{1}{2} in Corollary 6.9 yields

    \begin{equation*} \Gamma_X(V, W) = \tfrac{1}{2} \big(V W^{\top} + W V^{\top} \big) X + \tfrac{1}{2} X V^{\top} (I_n - X X^{\top}) W + \tfrac{1}{2} X W^{\top} (I_n - X X^{\top}) V \end{equation*}

    for X \in \mathrm{St}_{{{n}}, {{k}}} and V, W \in T_X \mathrm{St}_{{{n}}, {{k}}} . This expression coincides with [10, Eq. (2.49)].

    We investigated a multi-parameter family of metrics on an open U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}} such that \mathrm{St}_{{{n}}, {{k}}} \subseteq U becomes a pseudo-Riemannian submanifold. The corresponding geodesic equation for \mathrm{St}_{{{n}}, {{k}}} as explicit matrix-valued second order ODE was derived by computing the metric spray on T \mathrm{St}_{{{n}}, {{k}}} . In principle, this approach to determine the geodesic equation is not limited to \mathrm{St}_{{{n}}, {{k}}} . It seems to be applicable to a pseudo-Riemannian submanifold of an open subset of a vector space as soon as the metric spray on the open subset and the tangent map of the orthogonal projection are known. Beside the geodesic equation, several other quantities related to the geometry of the pseudo-Riemannian submanifold \mathrm{St}_{{{n}}, {{k}}} \subseteq U were determined in terms of explicit matrix-type formulas. In particular, the expressions for pseudo-Riemannian gradients and pseudo-Riemannian Hessians could pave the way for designing new optimization methods on \mathrm{St}_{{{n}}, {{k}}} . Moreover, we reproduced several well-known results from the literature putting them into a new perspective.

    This work has been supported by the German Federal Ministry of Education and Research (BMBF-Projekt 05M20WWA: Verbundprojekt 05M2020 - DyCA).

    The author declares there is no conflict of interest.



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