Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs

1.   Introduction

Numerical computations on the real (compact) Stiefel manifold viewed as the embedded submanifold $\mathrm{St}_{{{n}}, {{k}}} = \{X \in \mathbb{{{R}}}^{{{n}} \times {{k}}} \mid X^{\top} X = I_k \}$ of $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ 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 $\mathrm{St}_{{{n}}, {{k}}}$ 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 $\mathrm{St}_{{{n}}, {{k}}}$ discussed here.

In this paper, we investigate a $2k$-parameter family of pseudo-Riemannian metrics on $\mathrm{St}_{{{n}}, {{k}}}$ 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 $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ with a family of covariant $2$-tensors depending on $2k$ parameters, which are invariant under the $O(n)$-left action on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ 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 $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ such that $\mathrm{St}_{{{n}}, {{k}}} \subseteq U$ becomes a pseudo-Riemannian submanifold of $U$. Hence it makes sense to consider the normal bundle of $\mathrm{St}_{{{n}}, {{k}}}$ and the orthogonal projections onto the tangent spaces of $\mathrm{St}_{{{n}}, {{k}}}$ 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 $\mathrm{St}_{{{n}}, {{k}}}$, 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 $\mathrm{St}_{{{n}}, {{k}}}$ 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 $S \colon T \mathrm{St}_{{{n}}, {{k}}} \to T(T \mathrm{St}_{{{n}}, {{k}}})$ 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 $T U$, where $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ is the open set of which $\mathrm{St}_{{{n}}, {{k}}}$ 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 $T \mathrm{St}_{{{n}}, {{k}}}$ 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 $\mathrm{St}_{{{n}}, {{k}}}$ 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 $\mathrm{St}_{{{n}}, {{k}}}$ considered as pseudo-Riemannian submanifold of an open $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ 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.

2.   Terminology and notations

Throughout this text, except for Section 3.4, we view the real (compact) Stiefel manifold $\mathrm{St}_{{{n}}, {{k}}}$ as an embedded submanifold of the real $(n \times k)$-matrices $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ which is given by

We point out that $\mathrm{St}_{{{n}}, {{k}}}$ is a proper subset of $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ although the inclusion is denoted by $\mathrm{St}_{{{n}}, {{k}}} \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$. In the sequel, we often denote proper inclusions by "$\subseteq$". The symbol "$\subset$" is only used if we want to emphasize that an inclusion is not an equality. The tangent bundle of $\mathrm{St}_{{{n}}, {{k}}}$ is denoted by $T \mathrm{St}_{{{n}}, {{k}}}$ which is considered as a submanifold of $T \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}}$. More generally, for a manifold $M$, we denote by $T M$ and $T^* M$ its tangent and cotangent bundle, respectively. In the sequel, if not indicated other-wise, we identify $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ with its dual space $(\mathbb{{{R}}}^{{{n}} \times {{k}}})^*$ via the linear isomorphism

induced by the Frobenius scalar product. The following characterization of the tangent space of $\mathrm{St}_{{{n}}, {{k}}}$ at $X \in \mathrm{St}_{{{n}}, {{k}}}$ considered as subspace of $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ is used frequently

We write

for the orthogonal group and

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

Moreover, we write

for the projection onto $\mathfrak{{{so}}}(n)$ whose kernel is given by the set of symmetric matrices $\mathbb{{{R}}}^{{{n}} \times {{n}}}_{\mathrm{sym}}$. The $O(n)$-left action on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ by matrix multiplication from the left is denoted by

By restricting the second argument of $\Psi$ one obtains the $O(n)$-action

on $\mathrm{St}_{{{n}}, {{k}}}$ from the left which we denote by $\Psi$, as well. It is well-known that this $O(n)$-action on $\mathrm{St}_{{{n}}, {{k}}}$ is transitive. For fixed $R \in O(n)$ we denote the diffeomorphisms induced by the actions from (2.8) and (2.9)

both by $\Psi_R$.

If $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ is some subset, we write

for the canonical inclusion of $U$ into $\mathbb{{{R}}}^{{{n}} \times {{k}}}$. Moreover, the canonical inclusion of $\mathrm{St}_{{{n}}, {{k}}}$ into $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ is often denoted by

for short.

Next let ${\rm{pr}} \colon F \to M$ be a vector bundle over a manifold $M$ with dual bundle $F^*$. The smooth sections of $F$ are denoted by $\Gamma^{\infty}(F)$. Moreover, we denote by $F^{ \otimes \ell}$, $\mathrm{S}^\ell(F)$ and $\Lambda^\ell (F)$ the $\ell$-th tensor power, the $\ell$-th symmetrized tensor power and the $\ell$-th antisymmetrized tensor power of $F$, respectively. In addition, we write $\mathrm{End}(F) \cong F^* \otimes F$ for the endormorphism bundle of $F$. The vertical bundle of $F$ is denoted by $\mathrm{Ver}(F) \subseteq T F$.

Let $f \colon M \to N$ be a smooth map between manifolds and let $\alpha \in \Gamma^{\infty}\big((T^* N)^{ \otimes \ell} \big)$ be a covariant tensor field on $N$. The pullback of $\alpha$ by $f$ is denoted by $f^* \alpha$. If $\alpha$ is a differential form, i.e. $\alpha \in \Gamma^{\infty}\big(\Lambda^\ell (T^* M) \big)$, the exterior derivative of $\alpha$ is denoted by ${\rm{d}}\ \alpha$. The tangent map of $f$ is denoted by $T f \colon T M \to TN$. If $f$ is a map between (open subsets of) finite dimensional $\mathbb{{{R}}}$-vector spaces, we write ${\rm{D}} f(X) V$ 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 ${\rm{D}} f(X) V$, as well.

Next let $M \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ be a submanifold. A vector field $V \colon M \to TM \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}}$ is often implicitly identified with the map $M \to \mathbb{{{R}}}^{{{n}} \times {{k}}}$ defined by its second component which we denote by $V$, as well, i.e. the "foot point" $X \in M$ is suppressed in our notation. If $S \in \Gamma^{\infty}\big(T (TM) \big)$ is a vector field on $TM$, we view it as a map $S \colon TM \to T(T M) \subseteq \big(\mathbb{{{R}}}^{{{n}} \times {{k}}}\big)^4$ usually not suppressing the "foot point" $(X, V) \in TM$.

For a smooth function $F \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathbb{{{R}}}$ we write $\nabla F(X)$ for the gradient of $F$ at $X \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ with respect to the Frobenius scalar product, i.e. the unique matrix $\nabla F(X) \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ with

for all $V \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Furthermore $E_{ij} \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ denotes the matrix whose entries fulfill $(E_{ij})_{f \ell} = \delta_{i f} \delta_{j \ell}$ for all $f \in \{1, \ldots, n\}$ and $\ell \in \{1, \ldots, k\}$ with $\delta_{i f}$ and $\delta_{j \ell}$ 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.

3.   A family of metrics on the stiefel manifold

We start with investigating a $2k$-parameter family of symmetric covariant $2$- tensors on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$. For certain choices of these parameters, it defines a pseudo-Riemannian metric on an open subset $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ such that $\mathrm{St}_{{{n}}, {{k}}} \subseteq U$ becomes a pseudo-Riemannian submanifold of $U$.

3.1.   A symmetric $2$-Tensor on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ and its pull-back to $\mathrm{St}_{{{n}}, {{k}}}$

We introduce a $2k$-parameter family of symmetric covariant $2$-tensors on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$.

Lemma 3.1. Let $D = {\rm{diag}}(D_{11}, \ldots, D_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ and $E = {\rm{diag}}(E_{11}, \ldots, E_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ be both diagonal. Then the point-wise definition

with $X \in \mathbb{{{R}}}^{n \times k}$ and $V, W \in T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ yields a smooth covariant $2$-tensor $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathbb{{{R}}}^{{{n}} \times {{k}}}) \big)$ which is invariant under the $O(n)$-action $\Psi$ defined in (2.8).

Proof. Obviously, (3.1) defines a smooth covariant $2$-tensor. Let $R \in O(n)$. Then $\Psi_R^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} = \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ holds due to

for $X \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ and $V, W \in T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ showing the $\Psi$-invariance of $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$.

Remark 3.2. Observe that the diagonal entry $E_{ii} \in \mathbb{{{R}}}$ of the diagonal matrix $E = {\rm{diag}}(E_{11}, \ldots, E_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ shall not be confused with the matrix $E_{ii} \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ introduced at the end of Section 2. In the sequel, it should be clear by the context how the symbol $E_{ii}$ has to be understood.

Remark 3.3. Let $E = 0$. Then $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2(T^* \mathbb{{{R}}}^{n \times k})\big)$ becomes independent of $X \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Hence we may identify $\langle \cdot, \cdot \rangle^{D, 0}_{(\cdot)}$ with the symmetric bilinear form

If we want to emphasize that $\langle \cdot, \cdot \rangle^D$ is a symmetric bilinear form on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$, we denote it by $\langle \cdot, \cdot \rangle^D_{\mathbb{{{R}}}^{{{n}} \times {{k}}}}$.

Remark 3.4. The pull-back $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathrm{St}_{{{n}}, {{k}}})\big)$ of $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ with $\iota \colon \mathrm{St}_{{{n}}, {{k}}} \to \mathbb{{{R}}}^{{{n}} \times {{k}}}$ simplifies for the following values of $k$:

1. For $k = n$ one has $\mathrm{St}_{{{n}}, {{n}}} = O(n)$. Thus for $X \in O(n)$ and $V, W \in T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ one obtains

due to $X^{\top}X = X X^{\top} = I_n$, i.e. $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} = \langle \cdot, \cdot \rangle^{D + E}$ holds.

2. For $k = 1$ one has $\mathrm{St}_{{{n}}, {{1}}} = S^{n - 1} \subseteq \mathbb{{{R}}}^n$. Using $X^{\top} V = 0$ for all $X \in S^{n - 1}$ and $V \in T_X S^{n - 1}$ yields

i.e. $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} = \langle \cdot, \cdot \rangle^D$ holds.

Remark 3.5. The pull-back $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathrm{St}_{{{n}}, {{k}}}) \big)$ yields well-known metrics on $\mathrm{St}_{{{n}}, {{k}}}$ for certain choices of $D$ and $E$:

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

2. Setting $D = I_k$ and $E = - \tfrac{1}{2} I_k$ yields the canonical metric, see e.g. ^[10] or [18, Sec. 23.5]

3. For $D = 2 I_k$ and $E = \nu I_k$ with $\nu = - \tfrac{2 \alpha + 1}{\alpha + 1}$ and $\alpha \in \mathbb{{{R}}}\setminus \{- 1\}$ the metric $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ reproduces a one-parameter family which has been introduced in ^[13], see in particular [13, Eq. (55)].

In order to investigate $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathbb{{{R}}}^{{{n}} \times {{k}}}) \big)$ and its pull-back to $\mathrm{St}_{{{n}}, {{k}}}$ we first list some properties of $\langle \cdot, \cdot \rangle^D$.

Lemma 3.6. Let $D = {\rm{diag}}(D_{11}, \ldots, D_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ be diagonal. The following assertions are fulfilled:

1. The symmetric bilinear form $\langle \cdot, \cdot \rangle^D \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathbb{{{R}}}$ is a scalar product iff $D$ is invertible.

2. The bilinear form $\langle \cdot, \cdot \rangle^D \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathbb{{{R}}}$ is an inner product iff $D_{ii} > 0$ holds for all $i \in \{1, \ldots, k\}$.

3. Assume that $D$ is invertible. Then $\langle \cdot, \cdot \rangle^D \colon \mathbb{{{R}}}^{{{k}} \times {{k}}} \times \mathbb{{{R}}}^{{{k}} \times {{k}}} \to \mathbb{{{R}}}$ induces a scalar product on $\mathfrak{{{so}}}(k)$ iff

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

4. Let $k \geq 2$. Then $\langle \cdot, \cdot \rangle^D \big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}}\colon \mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k) \to \mathbb{{{R}}}$ defines an inner product on $\mathfrak{{{so}}}(n)$ iff

holds for all $1 \leq i < j \leq k$. For $k = 1$, this bilinear form defines always an inner product.

Proof. Let $E_{ij} \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ denote the matrix whose entries fulfill $(E_{ij})_{f \ell} = \delta_{i f} \delta_{j \ell}$. Clearly, the set

defines a basis of $\mathbb{{{R}}}^{{{n}} \times {{k}}}$. Thus it suffices to show that for all $E_{ij} \in B$ the associated linear forms

are non-zero iff $D$ is invertible. We have

with $V = (V_{ij}) \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Equation (3.8) implies that $D$ is invertible iff the linear forms in (3.7) are non-vanishing for all $i \in \{ 1, \ldots, n \}$ and $j \in \{ 1, \ldots, k \}$ showing Claim 1.

Next we prove Claim 2. Let $0 \neq V = (V_{ij}) \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Then $\langle V, V \rangle^D > 0$ holds iff $D_{ii} > 0$ for $i \in \{1, \ldots, k\}$ due to

We now prove Claim 3. For $k = 1$ the assertion is trivial due to $\dim(\mathfrak{{{so}}}(1)) = 0$. For $k \geq 2$ the set $\big\{ E_{ij} - E_{ji} \mid 1 \leq i < j \leq k \big\}$ is a basis of $\mathfrak{{{so}}}(k)$. Thus $\langle \cdot, \cdot \rangle^D$ induces a scalar product on $\mathfrak{{{so}}}(k)$ iff the linear forms

are non-vanishing for all $1 \leq i < j \leq k$. Writing $A = (A_{ij}) = (- A_{j i}) \in \mathfrak{{{so}}}(k)$ we compute

showing that $\langle \cdot, \cdot \rangle^D$ defines a scalar product on $\mathfrak{{{so}}}(k)$ iff

holds. Here we exploited that $D_{ii} + D_{ii} \neq 0$ is automatically fulfilled because $D$ is invertible.

It remains to prove Claim 4. The case $k = 1$ is trivial due to $\mathfrak{{{so}}}(1) = \{0\}$. Thus assume $k \geq 2$. Let $A = (A_{ij}) \in \mathfrak{{{so}}}(k)$. Exploiting $A_{ij} = - A_{ji}$ we calculate

Using $A_{ii} = 0$ we conclude that $\langle A, A \rangle^D > 0$ holds for all $0 \neq A \in \mathfrak{{{so}}}(k)$ iff $D_{ii} + D_{jj} > 0$ is fulfilled for all $1 \leq i < j \leq k$.

The next lemma shows that $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ induces a pseudo-Riemannian metric on the Stiefel manifold for certain choices of $D$ and $E$.

Lemma 3.7. Let $D = {\rm{diag}}(D_{11}. \ldots, D_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ and $E = {\rm{diag}}(E_{11}, \ldots, E_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ be both diagonal and let $X \in \mathrm{St}_{{{n}}, {{k}}}$. Then the following assertions are fulfilled:

1. Let $1 \leq k < n$. The bilinear form

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 $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ to $\mathrm{St}_{{{n}}, {{k}}}$ defines a pseudo-Riemannian metric on $\mathrm{St}_{{{n}}, {{k}}}$, i.e.

is a scalar product on $T_X \mathrm{St}_{{{n}}, {{k}}}$, iff the condition

holds.

3. Assume that (3.10) defines a scalar product. For $2 \leq k \leq n - 1$ the symmetric covariant $2$-tensor $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathrm{St}_{{{n}}, {{k}}})\big)$ is a Riemannian metric on $\mathrm{St}_{{{n}}, {{k}}}$, i.e.

is an inner product on $T_X \mathrm{St}_{{{n}}, {{k}}}$, iff the conditions $D_{ii} > 0$ for all $i \in \{1, \ldots, k\}$ and

are fulfilled. For $k = 1$ one obtains a Riemannian metric iff $D_{11} > 0$ holds. For $k = n$ the tensor $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ defines a Riemannian metric iff $D_{ii} + E_{ii} + D_{jj} + E_{jj} > 0$ holds for all $1 \leq i < j \leq n$.

Proof. Since the $O(n)$-left action $\Psi$ on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ defined in (2.8) is isometric with respect to $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ by Lemma 3.1 and, moreover, $\Psi$ restricts to a transitive action on $\mathrm{St}_{{{n}}, {{k}}}$ it suffices to prove the claims for a single point $X_0 \in \mathrm{St}_{{{n}}, {{k}}}$.

We first consider the case $k = n$. Then $\langle \cdot, \cdot \rangle_X^{D, E} = \langle \cdot, \cdot \rangle^{D + E}$ holds for all $X \in \mathrm{St}_{{{n}}, {{n}}} = O(n)$ by Remark 3.4, Claim 1. Hence $\langle \cdot, \cdot \rangle_X^{D, E}$ is non-degenerated iff $D + E$ is invertible according to Lemma 3.6, Claim 1. Next we consider the case $1 \leq k < n$. We choose $X_0 = I_{n, k}$, where

and write

with $V_1, W_1 \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ and $V_2, W_2 \in \mathbb{{{R}}}^{{{(n - k)}} \times {{k}}}$. By this notation and identifying $T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ we calculate

By (3.15) and Lemma 3.6, Claim 1, the bilinear form $\langle \cdot, \cdot \rangle_{I_{n, k}}^{D, E}$ defines a scalar product on $T_X \mathbb{{{R}}}^{{{n}} \times {{k}}}$ iff $D$ and $D + E$ are both invertible.

Next we assume that $D$ and $D + E$ are choosen such that $\langle \cdot, \cdot \rangle_{X}^{D, E}$ defines a scalar product on $T_{X} \mathbb{{{R}}}^{{{n}} \times {{k}}}$ for each $X \in \mathrm{St}_{{{n}}, {{k}}}$. We now prove Claim 2 for $1 \leq k \leq n - 1$. To this end, it is sufficient to show that

is a scalar product iff (3.12) holds. The tangent space $T_{I_{n, k}} \mathrm{St}_{{{n}}, {{k}}}$ is given by

see e.g. [10, Sec. 2.2.1]. Thus we may write $V, W \in T_X \mathrm{St}_{{{n}}, {{k}}}$ as

with $V_1, W_1 \in \mathfrak{{{so}}}(k)$ and $V_2, W_2 \in \mathbb{{{R}}}^{{{(n - k)}} \times {{k}}}$. We now obtain

analogously to (3.15). Clearly, Equation (3.18) defines a scalar product on $T_{I_{n, k}} \mathrm{St}_{{{n}}, {{k}}}$ iff

yields a scalar product on $\mathfrak{{{so}}}(k)$ and

defines a scalar product on $\mathbb{{{R}}}^{{{(n - k)}} \times {{k}}}$. 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 $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ and $T_{I_n} \mathrm{St}_{{{n}}, {{n}}} = \mathfrak{{{so}}}(n)$ as well as $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} = \langle \cdot, \cdot \rangle^{D + E}$ for $k = n$, Claim 2 follows by Lemma 3.6, Claim 3.

It remains to prove Claim 3. We first consider the case $2 \leq k \leq n - 1$. Since the bilinear form on $T_{I_{n, k}} \mathrm{St}_{{{n}}, {{k}}}$ induced by $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ 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 $\iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ is independent of $E$ due to $X^{\top} V = 0$ for all $X \in \mathrm{St}_{{{n}}, {{1}}}$ and $V \in T_X \mathrm{St}_{{{n}}, {{1}}}$, see also Remark 3.4, Claim 2. Hence (3.18) implies that $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ is positive definite iff

is positive definite. The desired result follows by Lemma 3.6, Claim 2. For $k = n$, the assertion holds due to $\langle \cdot, \cdot \rangle_X^{D, E} = \langle \cdot, \cdot \rangle^{D + E}$ for all $X \in \mathrm{St}_{{{n}}, {{n}}} = O(n)$ by Lemma 3.6, Claim 3.

The next lemma generalizing [14, Lem. 2] shows that there is an open neigbourhood $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ of $\mathrm{St}_{{{n}}, {{k}}}$ such that $\mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \iota_U^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big)$ is a pseudo-Riemannian submanifold. This fact is crucial for the following discussion.

Lemma 3.8. Let $D, E \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ be both diagonal such that for each $X \in \mathrm{St}_{{{n}}, {{k}}}$

defines a scalar product on $T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ which induces a scalar product on $T_X\mathrm{St}_{{{n}}, {{k}}} \subseteq T_X \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Then there exists an open neighbourhood $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ of $\mathrm{St}_{{{n}}, {{k}}}$ such that $\iota_U^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* U) \big)$ is a pseudo-Riemannian metric on $U$ and $\big(\mathrm{St}_{{{n}}, {{k}}}, \iota^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big)$ is a pseudo-Riemannian submanifold of $\big(U, \iota_U^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big)$.

Proof. We identify $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathbb{{{R}}}^{{{n}} \times {{k}}})\big)$ with the continuous map

The bilinear form $\varphi(X) = \langle \cdot, \cdot \rangle_X^{D, E} \in \mathrm{S}^2\big((\mathbb{{{R}}}^{{{n}} \times {{k}}})^* \big)$ is a scalar product for all $X \in \mathrm{St}_{{{n}}, {{k}}}$ by assumption. Hence, by the continuity of $\varphi$, there is an on open neighbourhood $U_X$ of $X$ in $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ such that $\varphi(\widetilde{X}) \in \mathrm{S}^2\big((\mathbb{{{R}}}^{{{n}} \times {{k}}})^*\big)$ is non-degnerated for all $\widetilde{X} \in U_X$. We set

Then $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ is open as a union of open sets and fulfills $\mathrm{St}_{{{n}}, {{k}}} \subseteq U$ by definition. Moreover, $\varphi(\widetilde{X})$ is non-dengenerated for all $\widetilde{X} \in U$ by construction. Hence $\iota_U^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ defines a pseudo-Riemannian metric on $U$ such that $\mathrm{St}_{{{n}}, {{k}}} \subseteq \big(U, \iota_U^* \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big)$ is a pseudo-Riemannian submanifold.

Obviously, the inclusion $\mathrm{St}_{{{n}}, {{k}}} \subseteq U$ from Lemma 3.8 is always proper since $\mathrm{St}_{{{n}}, {{k}}}$ is closed in $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ while $U$ is open in $\mathbb{{{R}}}^{{{n}} \times {{k}}}$.

Notation 3.9. From now on, unless indicated otherwise, pull-backs of $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ to submanifolds of $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ are suppressed in the notation.

3.2.   Number of parameters

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

Lemma 3.10. Let $D = {\rm{diag}}(D_{11}, \ldots, D_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ be some diagonal matrix. Then the following assertions are fulfilled:

1. The bilinear form $\langle \cdot, \cdot \rangle^D \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathbb{{{R}}}$ vanishes identically iff $D = 0$ holds.

2. The restriction $\langle \cdot, \cdot \rangle^D\big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}}\colon \mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k) \to \mathbb{{{R}}}$ of $\langle \cdot, \cdot \rangle^D \colon \mathbb{{{R}}}^{{{k}} \times {{k}}} \times \mathbb{{{R}}}^{{{k}} \times {{k}}} \to \mathbb{{{R}}}$ fulfills the following assertions:

(a) For $k = 1$ one has $\langle \cdot, \cdot \rangle^D\big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}} = 0$ for all $D \in \mathbb{{{R}}}^{{{1}} \times {{1}}} \cong \mathbb{{{R}}}$.

(b) For $k = 2$ one has $\langle \cdot, \cdot \rangle^D\big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}} = 0$ iff $D_{11} + D_{22} = 0$ holds.

(c) For $k \geq 3$ one has $\langle \cdot, \cdot \rangle^D\big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}} = 0$ iff $D = 0$ holds.

Proof. Let $E_{ij} \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ the matrix whose entries fulfill $(E_{ij})_{f \ell} = \delta_{i f} \delta_{j \ell}$. Then

where $V = (V_{ij}) \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Since $\langle \cdot, \cdot \rangle^D = 0$ holds iff the linear forms $\langle E_{ij}, \cdot \rangle^D \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathbb{{{R}}}$ vanishes for all $1 \leq i \leq n$ and $1 \leq j \leq k$, the first claim follows by (3.20).

Next, we consider $\langle \cdot, \cdot \rangle^D\big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}}\colon \mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k) \to \mathbb{{{R}}}$. Clearly, it vanishes for $k = 1$ for all $D \in \mathbb{{{R}}}^{{{1}} \times {{1}}}$ due to $\mathfrak{{{so}}}(1) = \{0\}$.

We now assume $k \geq 2$. Then $\langle \cdot, \cdot \rangle^D\big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}}\colon \mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k) \to \mathbb{{{R}}}$ vanishes iff the linear forms

vanish for all $1 \leq i < j \leq k$. Writing $A = (A_{ij}) = (-A_{ji}) \in \mathfrak{{{so}}}(k)$ we obtain

Thus the linear forms (3.21) are zero iff $D_{ii} + D_{jj} = 0$ holds for all $1 \leq i < j \leq k$. For $k = 2$ this is equivalent to $D_{11} + D_{22} = 0$. It remains to consider the case $k \geq 3$. The conditions $D_{ii} + D_{jj} = 0$ for all $1 \leq i < j \leq k$ include the conditions

and

In particular $D_{11} = - D_{k - 1}$ and $D_{11} = -D_{kk}$ holds. Plugging these identities into (3.23) yields

Hence (3.22) implies $D_{ii} = 0$ for all $2 \leq i \leq k$. Therefore $\langle \cdot, \cdot \rangle^D\big\vert_{{\mathfrak{{{so}}}(k) \times \mathfrak{{{so}}}(k)}} = 0$ iff $D = 0$ as desired.

The next lemma justifies calling $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ a $2k$-parameter family provided that $3 \leq k \leq n - 1$ holds.

Lemma 3.11. Let

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

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

1. For $k = 1 = n$, one has $\dim\big({\rm{im}}(\psi) \big) = 0$ and $\ker(\psi) = \mathbb{{{R}}}\times \mathbb{{{R}}}$.

2. For $k = 1$ and $n > 1$ one has $\dim\big({\rm{im}}(\psi)\big) = 1$ and $\ker(\psi) = \{(0, E) \mid E \in \mathbb{{{R}}}\} \subseteq \mathbb{{{R}}}\times \mathbb{{{R}}}$.

3. For $k = 2 = n$ one has $\dim\big({\rm{im}}(\psi) \big) = 1$ and

4. For $2 < k < n$ one has $\dim\big({\rm{im}}(\psi) \big) = 2k$ and $\ker(\psi) = \{0\} \subseteq \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \times \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}}$.

5. For $k = n > 2$ one has $\dim\big({\rm{im}}(\psi)\big) = k$ and $\ker(\psi) = \{(D, - D) \mid D \in \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \} \subseteq \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \times \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}}$.

Proof. Clearly, the map $\psi$ is linear. Next we define the linear map

Obviously, for each $(D, E) \in \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \times \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}}$ one has $\big(\psi(D, E)\big)({I_{n, k}}) = \langle \cdot, \cdot \rangle_{I_{n, k}}^{D, E} = \widetilde{\psi}(D, E)$. Since $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ is invariant under the transitive $O(n)$-action $\Psi$ on $\mathrm{St}_{{{n}}, {{k}}}$ according to Lemma 3.1, this yields

Moreover, the equivalence

is clearly fulfilled. We again write

with $V_1, W_1 \in \mathfrak{{{so}}}(k)$ and $V_2, W_2 \in \mathbb{{{R}}}^{{{(n - k)}} \times {{k}}}$. By this notation and the description of $\ker(\widetilde{\psi})$ from (3.26), we study each case separately:

1. Obviously, for $k = 1 = n$ the claim $\ker(\widetilde{\psi}) = \mathbb{{{R}}} \times \mathbb{{{R}}}$ is correct due to $T_{I_{1}} \mathrm{St}_{{{1}}, {{1}}} = \{0\}$ implying $\dim\big(S^2 (T^*_{I_{1}} \mathrm{St}_{{{1}}, {{1}}})\big) = 0$.

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

Clearly, Equation (3.27) vanishes iff $D = 0$ holds independent of the value of $D + E$ by Lemma 3.10. Hence the kernel of $\psi$ is given by $\ker(\widetilde{\psi}) = \{ (0, E) \mid E \in \mathbb{{{R}}} \}$

3. For $k = 2 = n$ we have

Lemma 3.10 yields $\widetilde{\psi}(D, E) = 0$ iff $(D + E)_{11} + (D + E)_{22} = 0$ is fulfilled. Therefore we obtain

4. We now consider the case $3 \leq k \leq n - 1$. Then one has

By Lemma 3.10, we have $\widetilde{\psi}(D, E) = 0$ iff $D = 0$ and $D + E = 0$ holds. Therefore the kernel of $\psi$ is given by $\ker(\widetilde{\psi}) = \big\{ (D, E) \in \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \times \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \mid D = 0 = E \big\} = \{0\}$.

5. It remains to consider the case $k = n \geq 3$. We obtain

for all $V, W \in T_{I_n} \mathrm{St}_{{{n}}, {{n}}} = \mathfrak{{{so}}}(n)$. Thus $\widetilde{\psi}(D, E) = 0$ holds iff $D + E = 0$ is fulfilled by Lemma 3.10. Hence the kernel of $\widetilde{\psi}$ is given by $\ker(\widetilde{\psi}) = \big\{ (D, - D) \mid D \in \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \big\}$.

The equality $\ker(\psi) = \ker(\widetilde{\psi})$ is satisfied according to (3.25). Moreover, we have

as desired.

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

contains the non-empty subset $\big\{(D, E) \in \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \times \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \mid D_{ii} > 0 \text{ and } E_{ii} > 0 \text{ for all } i \in \{1, \ldots, k\} \big\}$ which is open in $\mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \times \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}}$. Moreover, the linear map $\psi \colon \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \times \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}} \to \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathrm{St}_{{{n}}, {{k}}})\big)$ is injective for $2 < k < n$ according to Lemma 3.11. This point of view justifies calling $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ a $2k$-parameter family at least for $2 < k < n$. For other choices of $k$ and $n$ one has rather a $\big(\! \dim \! \big(\! {\rm{im}}(\psi)\big)\big)$-parameter family of metrics. However, ignoring this over parameterization, we call them $2k$-parameter family, nevertheless.

3.3.   Orthogonal projections onto tangent spaces

The Stiefel manifold $\mathrm{St}_{{{n}}, {{k}}}$ endowed with $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathrm{St}_{{{n}}, {{k}}})\big)$ can be viewed as a pseudo-Riemannian submanifold of $\big(U, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big)$ with some suitable open $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ by Lemma 3.8. Consequently, for any given point $X \in \mathrm{St}_{{{n}}, {{k}}}$, we may consider the orthogonal projection

where $T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ is endowed with the scalar product $\langle \cdot, \cdot \rangle_X^{D, E}$. Moreover, it makes sense to consider the normal space $N_X \mathrm{St}_{{{n}}, {{k}}} = \big(T_X \mathrm{St}_{{{n}}, {{k}}} \big)^{\perp} \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ with respect to $\langle \cdot, \cdot \rangle_X^{D, E} \colon T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \times T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathbb{{{R}}}$.

Notation 3.13. From now on, unless indicated otherwise, we always assume that $D, E \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ are both diagonal matrices such that $\langle \cdot, \cdot \rangle_X^{D, E}$ defines a scalar product on $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ for each $X \in \mathrm{St}_{{{n}}, {{k}}}$ and $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ induces a pseudo-Riemannian metric on $\mathrm{St}_{{{n}}, {{k}}}$. 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 $D = {\rm{diag}}(D_{11}, \ldots, D_{kk}) \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ be invertible such that $D_{ii} + D_{jj} \neq 0$ holds for all $i, j \in \{1, \ldots, k\}$. Then the following assertions are fulfilled:

1. The orthogonal complement of $\mathfrak{{{so}}}(k)$ in $\mathbb{{{R}}}^{{{k}} \times {{k}}}$ with respect to the scalar product $\langle \cdot, \cdot \rangle^D$ is given by

Moreover, $\mathfrak{{{so}}}(k) \oplus \mathfrak{{{so}}}(k)^{\perp_D} = \mathbb{{{R}}}^{{{k}} \times {{k}}}$ holds.

2. The orthogonal projection

onto $\mathfrak{{{so}}}(k)$ with respect $\langle \cdot, \cdot \rangle^D$ is entry-wise given by

Proof. We first determine $\mathfrak{{{so}}}(k)^{\perp_D}$. To this end, we calculate

Let $\Lambda \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}}$. Then $(\Lambda D^{-1}) D = \Lambda = \Lambda^{\top} = D (\Lambda D^{-1})^{\top}$ showing $\{ \Lambda D^{-1} \mid \Lambda \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}} \} \subseteq \mathfrak{{{so}}}(k)^{\perp_D}$. The equality $\mathfrak{{{so}}}(k)^{\perp_D} = \big\{ \Lambda D^{-1} \mid \Lambda \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}} \big\}$ follows by counting dimensions. By Lemma 3.6, Claim 3 the assumptions on $D$ ensure that $\langle \cdot, \cdot \rangle^D$ induces a scalar product on $\mathfrak{{{so}}}(k)$. Hence $\mathfrak{{{so}}}(k) \oplus \mathfrak{{{so}}}(k)^{\perp_D} = \mathbb{{{R}}}^{{{k}} \times {{k}}}$ holds, see e.g. [17, Chap. 2, Lem. 23].

It remains to prove Claim 2. To this end, we show ${\rm{im}}({\pi^D}) = \mathfrak{{{so}}}(k)$ and $\ker(\pi^D) = \mathfrak{{{so}}}(k)^{\perp_D}$ as well as $\pi^D\big\vert_{{\mathfrak{{{so}}}(k)}} = \mathrm{id}_{\mathfrak{{{so}}}(k)}$. We first prove ${\rm{im}}(\pi^D) \subseteq \mathfrak{{{so}}}(n)$. Let $A = (A_{ij}) \in \mathbb{{{R}}}^{k \times k}$. We compute

for $i, j \in \{1, \ldots, k\}$ showing ${\rm{im}}(\pi^D) \subseteq \mathfrak{{{so}}}(k)$. Moreover, for $A \in \mathfrak{{{so}}}(k)$, i.e. $A_{ij} = - A_{ji}$, we have

This yields $\pi^D(A) = A$ for all $A \in \mathfrak{{{so}}}(k)$, i.e. $\pi^D\big\vert_{{\mathfrak{{{so}}}(k)}} = \mathrm{id}_{\mathfrak{{{so}}}(k)}$. Moreover, the inclusion ${\rm{im}}(\pi^D) \subseteq \mathfrak{{{so}}}(k)$ is in fact an equality. Next let $A \in \mathfrak{{{so}}}(k)^{\perp_D}$. Then $AD = D A^{\top}$ holds according to Claim 1 implying

Thus $\pi^D \big\vert_{{\mathfrak{{{so}}}^{\perp_D}}} = 0$ follows.

The formula for $\pi^D$ can be rewritten in terms of the so-called Hadamard or Schur product. For matrices $A, B \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$, it is entry-wise defined by

Remark 3.15. Let $\mu \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ be defined entry-wise by

Then the projection $\pi^D \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to \mathfrak{{{so}}}(k)$ from Lemma 3.14 can be rewritten as

Corollary 3.16. Let $0 \neq \beta \in \mathbb{{{R}}}$ and define $D = \beta I_k$. Then, for each $A \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ the map $\pi^D$ from Lemma 3.14 simplifies to

Proof. The desired result follows by a straightforward calculation exploiting $D_{ii} = \beta \neq 0$ for all $i \in \{1, \ldots, k\}$.

We determine the normal spaces of $\mathrm{St}_{{{n}}, {{k}}}$ with respect $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ generalizing [19, Chap. 1, Lem. 3.15] and [14, Lem. 3].

Lemma 3.17. The normal space $N_X \mathrm{St}_{{{n}}, {{k}}} = \big(T_X \mathrm{St}_{{{n}}, {{k}}} \big)^\perp \subseteq T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ at $X \in \mathrm{St}_{{{n}}, {{k}}}$ with respect to $\langle \cdot, \cdot \rangle_X^{D, E}$ is given by

Proof. Clearly, the set $\big\{ X \Lambda (D + E)^{-1} \in \mathbb{{{R}}}^{{{n}} \times {{k}}} \mid \Lambda = \Lambda^{\top} \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}} \big\}$ is a linear subspace of $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ of dimension $(k^2 + k) /2$ being the image of the injective linear map

Moreover, every matrix $V = X \Lambda (D + E)^{-1}$ with $\Lambda \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}}$ is orthogonal to the tangent space $T_X \mathrm{St}_{{{n}}, {{k}}}$. Indeed, we have for $W \in T_X \mathrm{St}_{{{n}}, {{k}}}$

due to $\Lambda = \Lambda^{\top}$ and $X^{\top} W = - W^{\top} X$. Therefore $\big\{ X \Lambda (D + E)^{-1} \in \mathbb{{{R}}}^{{{n}} \times {{k}}} \mid \Lambda = \Lambda^{\top} \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}} \big\} \subseteq N_X \mathrm{St}_{{{n}}, {{k}}}$ follows. By counting dimensions, this inclusion is in fact an equality.

Theorem 3.18. Let $X \in \mathrm{St}_{{{n}}, {{k}}}$. The orthogonal projection of $T_X \mathbb{{{R}}}^{{{n}} \times {{k}}} \cong \mathbb{{{R}}}^{{{n}} \times {{k}}}$ onto $T_X \mathrm{St}_{{{n}}, {{k}}} \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ with respect to $\langle \cdot, \cdot \rangle_X^{D, E}$ is given by

Proof. We first show ${\rm{im}}(P_X) = T_X \mathrm{St}_{{{n}}, {{k}}}$. Let $X \in \mathrm{St}_{{{n}}, {{k}}}$ and $V \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$. One calculates

Moreover, using ${\rm{im}}(\pi^{D + E}) = \mathfrak{{{so}}}(n)$, we obtain

Hence $X^{\top} \big(P_X(V) \big) = \pi^{D + E}(X^{\top} V) = - \big(P_X(V) \big)^{\top} X$ follows, i.e. ${\rm{im}}(P_X) \subseteq T_X \mathrm{St}_{{{n}}, {{k}}}$ as desired.

We now assume $V \in T_X \mathrm{St}_{{{n}}, {{k}}}$. By using $X^{\top} V = - V^{\top} X$ and $\pi^D\big\vert_{{\mathfrak{{{so}}}(n)}} = \mathrm{id}_{\mathfrak{{{so}}}(n)}$, we calculate

proving $P_X\big\vert_{{T_X \mathrm{St}_{{{n}}, {{k}}}}} = \mathrm{id}_{T_X \mathrm{St}_{{{n}}, {{k}}}}$ and implying that ${\rm{im}}(P_X) \subseteq T_X \mathrm{St}_{{{n}}, {{k}}}$ is indeed an equality.

It remains to show $\ker(P_X) = (T_X \mathrm{St}_{{{n}}, {{k}}})^{\perp}$. Let $V \in N_X \mathrm{St}_{{{n}}, {{k}}}$. We may write $V = X \Lambda (D + E)^{-1}$ with some suitable symmetric matrix $\Lambda \in \mathbb{{{R}}}^{{{k}} \times {{k}}}_{\mathrm{sym}}$ by exploiting Lemma 3.17. Consequently, we have

by using Lemma 3.14, Claim 1 which shows $\pi^{D + E}\big(\Lambda (D + E)^{-1} \big) = 0$.

Theorem 3.18 reproduces several results known in the literature.

Remark 3.19. Let $X \in \mathrm{St}_{{{n}}, {{k}}}$. We obtain the following special cases for $P_X \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \to T_X \mathrm{St}_{{{n}}, {{k}}}$ by using Corollary 3.16:

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

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

2. More generally, for $D = 2 I_k$ and $E = \nu I_n$ with $\nu \in \mathbb{{{R}}} \setminus \{-2\}$ one obtains

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

Next we determine an orthonormal basis of $\big(T_{I_{n, k}} \mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}\big)$ which allows for computing the signature of $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$, as well.

Remark 3.20. We define the subsets $B_1, B_2 \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ such that $B = B_1 \cup B_2$ is an orthonormal basis of $\big(T_{I_{n, k}} \mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle^{D, E}_{I_{n, k}}\big)$. To this end, let $E_{ij} \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ denote the matrix whose entries fulfill $(E_{ij})_{f \ell} = \delta_{i f} \delta_{j \ell}$ as usual. We set $B_1 = \emptyset$ for $k = 1$ and define

Moreover, we set

and $B_2 = \emptyset$ for $k = n$. A straightforward calculation shows $\langle V, W \rangle_{I_{n, k}}^{D, E} = 0$ for all $V, W \in B$ with $V \neq W$. Moreover, for $V = W \in B$ one obtains

and

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

where $\sharp S$ denotes the number of elements in the finite set $S$.

3.4.   Stiefel manifolds as reductive homogeneous spaces

Before we continue with the extrinsic approach, we briefly discuss the metric $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ on $\mathrm{St}_{{{n}}, {{k}}}$ viewed as a pseudo-Riemannian reductive homogeneous $SO(n)$-space. This point of view allows for relating $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ 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 $1 \leq k \leq n - 1$ and $n \geq 3$. Then the Killing form on $SO(n)$ given by

is negative definite, see e.g. [18, Sec. 21.6]. In addition, $\mathrm{St}_{{{n}}, {{k}}}$ is diffeomorphic to the reductive homogeneous space $SO(n) / SO(n - k)$, where $SO(n - k)$ is realized as a closed subgroup of $SO(n)$ via

and a reductive split is given by $\mathfrak{{{so}}}(n) = \mathfrak{{{h}}} \oplus \mathfrak{{{m}}}$, where

see e.g. [18, Sec. 23.5]. In particular, since $SO(n) \times \mathrm{St}_{{{n}}, {{k}}} \ni (R, X) \mapsto R X \in \mathrm{St}_{{{n}}, {{k}}}$ is a transitive $SO(n)$-left action whose stabilizer subgroup of $I_{n, k}$ coincides with $SO(n - k) \subseteq SO(n)$, the map

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

Here $R \cdot SO(n - k) \in SO(n) / SO(n - k)$ denotes the coset defined by $R \in SO(n)$. 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

on $\mathfrak{{{so}}}(n)$ which induces a left-invariant metric on $SO(n)$ such that (3.44) becomes a pseudo-Riemannian submersion. In addition, equipping $SO(n) / SO(n - k)$ with this submersion metric turns (3.45) into a $SO(n)$-equivariant isometry to $\big(\mathrm{St}_{{{n}}, {{k}}}, \langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \big)$.

Throughout this section we denote by $D, E \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ diagonal matrices such that $D$ and $D + E$ are both invertible, see also Notation 3.13.

Lemma 3.21. Let $E_{ij} \in \mathbb{{{R}}}^{{{n}} \times {{n}}}$ be the matrix whose entries fulfill $(E_{ij})_{f \ell} = \delta_{i f} \delta_{j \ell}$ and let $F\! = \! D + E \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$. Then

is linear, where $\xi_{11} \in \mathfrak{{{so}}}(k)$, $\xi_{22} \in \mathfrak{{{so}}}(n - k)$ and $\xi_{21} \in \mathbb{{{R}}}^{{{(n - k)}} \times {{k}}}$. Moreover, evaluating $A$ at the basis $\big\{ (E_{ij} - E_{ji}) \mid 1 \leq i < j \leq k \big\}$ of $\mathfrak{{{so}}}(n)$ yields

In particular, $A \colon \mathfrak{{{so}}}(n) \to \mathfrak{{{so}}}(n)$ as well as its restriction $A\big\vert_{{\mathfrak{{{m}}}}}\colon \mathfrak{{{m}}} \to \mathfrak{{{m}}}$ 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 $1 \leq i < j \leq k$. Then $E_{ij} - E_{ji}$ is mapped by $A$ to

with $\widehat{E}_{ij} \in \mathbb{{{R}}}^{{{k}} \times {{k}}}$ defined by $\big(\widehat{E}_{ij}\big)_{f \ell} = \delta_{i f} \delta_{j \ell}$. Next assume $k + 1 \leq i \leq n$ and $1 \leq j \leq k$. One obtains

The equality $A\big(E_{ij} - E_{ji} \big) = E_{ij} - E_{ji}$ for $k + 1 \leq i < j \leq n$ is obvious.

Lemma 3.22. Define

where $A \colon \mathfrak{{{so}}}(n) \to \mathfrak{{{so}}}(n)$ is the linear map from Lemma 3.21. Then the following assertions are fulfilled:

1. $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ is a scalar product on $\mathfrak{{{so}}}(n)$.

2. The restriction of $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ to $\mathfrak{{{m}}}$ defines a scalar product $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)} \colon \mathfrak{{{m}}} \times \mathfrak{{{m}}} \to \mathbb{{{R}}}$ on $\mathfrak{{{m}}}$.

3. Writing

with $\xi_{11}, \eta_{11} \in \mathfrak{{{so}}}(k)$ and $\xi_{21}, \eta_{21} \in \mathbb{{{R}}}^{{{(n - k)}} \times {{k}}}$ yields

4. $\langle \cdot, \cdot \rangle^{red(D, E)}$ is ${\rm{Ad}}(SO(k))$-invariant.

5. Declaring $T_e {\rm{pr}} \colon T_e SO(n) \to T_{{\rm{pr}}(e)} \big(SO(n) / SO(n - k) \big)$ as an isometry defines a $SO(n)$-invariant pseudo-Riemannian metric on $SO(n) / SO(n - k)$ such that ${\rm{pr}} \colon SO(n) \to SO(n) / SO(n - k)$ is a pseudo-Riemannian submersion, where $SO(n)$ is equipped with the left-invariant metric defined by $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$.

Proof. Obviously, $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ is a bilinear form. Using the notation introduced in (3.49) one calculates

Hence $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ is symmetric. Claim 3 follows by (3.51), as well. Moreover, $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ is a scalar product since $A \colon \mathfrak{{{so}}}(n) \to \mathfrak{{{so}}}(n)$ is a linear isomorphism by Lemma 3.21 showing Claim 1. Claim 2 follows since $A \big\vert_{{m}}\colon \mathfrak{{{m}}} \to \mathfrak{{{m}}}$ is an isomorphism, too.

In order to show the ${\rm{Ad}}\big(SO(n - k)\big)$-invariance we calculate

for $\xi = \left[ \begin{matrix} x_{11} & & - \xi_{21}^{\top} \\ \xi_{21} & & \xi_{22} \end{matrix} \right] \in \mathfrak{{{so}}}(n)$ and $g = \left[ \begin{matrix} I_k & & 0 \\ 0 & & R \end{matrix} \right] \in SO(n - k) \subseteq SO(n)$ implying

as desired.

It remains to prove Claim 5. By (3.50) the vector spaces $\mathfrak{{{m}}} \subseteq \mathfrak{{{so}}}(n)$ and $\mathfrak{{{h}}} \subseteq \mathfrak{{{so}}}(n)$ are orthogonal complements with respect to $\langle \cdot, \cdot \rangle^{\mathrm{red(D, E)}}$. Moreover, by exploiting the ${\rm{Ad}}\big(SO(n- k)\big)$-invariance of $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$, 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 $\langle \cdot, \cdot \rangle^{\mathrm{red(D, E)}}$ has indeed the desired property. To this end, the tangent map of (3.44) at $I_n \in SO(n)$ is determined as

Proposition 3.23. Let $SO(n) / SO(n - k)$ be equipped with the pseudo-Riemannian metric constructed by means of the scalar product $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)} \colon \mathfrak{{{m}}} \times \mathfrak{{{m}}} \to \mathbb{{{R}}}$ and let $\mathrm{St}_{{{n}}, {{k}}}$ be endowed with the metric $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$.

1. The restriction of (3.52) to $\mathfrak{{{m}}}$, i.e. the linear map

is an isometry, where $T_{I_{n, k}} \mathrm{St}_{{{n}}, {{k}}}$ is equipped with the scalar product $\langle \cdot, \cdot \rangle_{I_{n, k}}^{D, E}$.

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

Proof. We write $\xi, \eta \in \mathfrak{{{m}}}$ as

with $\xi_{11}, \eta_{11} \in \mathfrak{{{so}}}(k)$ as well as $\xi_{21}, \eta_{21} \in \mathbb{{{R}}}^{{{(n -k)}} \times {{k}}}$ and compute

where the last equality holds by Lemma 3.22, Claim 3. It remains to show Claim 2. Since the metric on $SO(n) / SO(n - k)$ induced by $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ and the metric $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ on $\mathrm{St}_{{{n}}, {{k}}}$ are both $SO(n)$-invariant, the map $\check{{\rm{pr}}} \colon SO(n) / SO(n- k) \to \mathrm{St}_{{{n}}, {{k}}}$ is an isometry by Claim 1 due to its $SO(n)$-equivariance.

Proposition 3.23 allows for relating the metric $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E} \in \Gamma^{\infty}\big(\mathrm{S}^2 (T^* \mathrm{St}_{{{n}}, {{k}}}) \big)$ to the metrics on $\mathrm{St}_{{{n}}, {{k}}}$ defined in [15, Eq. (3.2)]. In order to compare these metrics we introduce some notation following ^[15]. We choose $k_1, \ldots, k_s \in \mathbb{{{N}}}$ with

and write

where $\widetilde{D}_{ii}, \widetilde{E}_{ii} \in \mathbb{{{R}}}$. Using the notation from ^[15] with $\mathfrak{{{p}}} = \mathfrak{{{m}}}$ and $t = 1$ we rewrite $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ as

Here $\langle \cdot, \cdot \rangle_{\mathfrak{{{p}}}_{ij}} = (n - 2) {\rm{tr}}\big((\cdot)^{\top} (\cdot) \big)\big\vert_{{\mathfrak{{{p}}}_{ij} \times \mathfrak{{{p}}}_{ij}}}$ denotes the Killing form on $\mathfrak{{{so}}}(n)$ scaled by $-1$ restricted to $\mathfrak{{{p}}}_{ij}$. Hence $\langle \cdot, \cdot \rangle^{\mathrm{red}(D, E)}$ coincides with the inner product defined in [15, Eq. (3.2)], where $x_{i} = x_{ii}$ and

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

holds. This can be seen by observing that for $\xi \in \mathfrak{{{m}}} = \mathfrak{{{p}}}$ the unique decomposition of $\xi$ into sums of $\xi_{ij} \in \mathfrak{{{p}}}_{ij}$ 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 $E$

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

and therefore $y = \tfrac{(n - 2)(2 z + (x - 2 z) + 2 z + (x - 2z))}{2 (n - 2)} = x$ holds for $x, z \in \mathbb{{{R}}}$. In particular, the metrics on $\mathrm{St}_{{{n}}, {{k}}}$ defined by $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ contain only the two $SO(n) \times SO(k)$-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 $\mathrm{St}_{{{n}}, {{k}}}$ defined by $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$, we are not able to rule out that the family $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ includes Einstein metrics different from the Jensen metrics. However, searching for Einstein metrics in $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ is out of the scope of this text.

4.   Sprays and geodesic equations

The goal of this section is to derive an explicit expression for the spray $S \in \Gamma^{\infty}\big(T (T \mathrm{St}_{{{n}}, {{k}}}) \big)$ associated with the metric $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$. An expression for $S$ yields an expression for the geodesic equation with respect to $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ 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 $\big(M, \langle \cdot, \cdot \rangle \big)$ be a pseudo-Riemannian manifold. The metric spray $S \in \Gamma^{\infty}\big(T (T M)\big)$ is the unique spray which is associated with the Levi-Civita covariant derivative defined by the pseudo-Riemannian metric $\langle \cdot, \cdot \rangle$.

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 $\big(M, \langle \cdot, \cdot \rangle\big)$ be pseudo-Riemannian and let $\omega_0 \in \Gamma^{\infty}\big(\Lambda^2T^* (T^* M) \big)$ denote the canonical symplectic form on $T^* M$. It is given by

with $\theta_0 \in \Gamma^{\infty}\big(T^* (T^* M)\big)$ being the canonical $1$-form on $T^* M$. We refer to [16, Sec. 6.2] for the definition of $\omega_0$ and $\theta_0$. Consider the Lagrange function

Let ${\rm{F}} L \colon T M \to T^* M$ denote the fiber derivative of $L$ defined by

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

is a closed $2$-from on $TM$, the so-called Lagrangian $2$-form, see [16, Sec. 7.2]. In addition, $\omega_L$ is non-degenerated, i.e. symplectic, since ${\rm{F}} L \colon T M \to T^* M$ is a diffeomorphism due to

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

associated with $L$ fulfills $E_L = L$, see e.g. [16, Sec. 7.3]. Let $X_{E_L} \in \Gamma^{\infty}\big(T (TM) \big)$ denote the Lagrangian vector field and write $i_{X_{E_L}} \omega_L$ for the insertion of $X_{E_L}$ into the first argument of $\omega_L$ as usual. Then $X_{E_L}$ is uniquely determined by

according to [16, Sec. 7.3]. Moreover, the Lagrangian vector field $X_{E_L}$ coincides with the spray associated with the metric $\langle \cdot, \cdot \rangle$, 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 $S \colon TM \to T(TM)$ can be represented in a chart $\big(TU, (x, v)\big)$ of $TM$ induced by a chart $(U, x)$ of $M$ by

Here $\Gamma_x$ denotes the quadratic map defined by $\big(\Gamma_x(v, v)\big)^k = \Gamma_{ij}^k(x) v^i v^j$ using Einstein summation convention, where $\Gamma_{ij}^k$ 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 $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ denotes an open subset of $\mathbb{{{R}}}^{{{n}} \times {{k}}}$ with the property from Lemma 3.8. Moreover, we denote by $\widetilde{L}$ the Lagrange function

where we identify $T U \cong U \times \mathbb{{{R}}}^{{{n}} \times {{k}}}$ as usual.

We use the formula for $\omega_0 \in \Gamma^{\infty}\big(\Lambda^2 T^* (T^* U) \big)$ on $T^*U$ given in the next remark.

Remark 4.3. The canonical symplectic form $\omega_0 \in \Gamma^{\infty}\big(\Lambda^2 T^* (T^* U) \big)$ on $T^* U$ is given by

for $(X, V, Y, Z), (X, V, \widetilde{Y}, \widetilde{Z}) \in T(T^* U)$ identifying $T (T^* U) \cong U \times (\mathbb{{{R}}}^{{{n}} \times {{k}}})^* \times \mathbb{{{R}}}^{{{n}} \times {{k}}} \times (\mathbb{{{R}}}^{{{n}} \times {{k}}})^*$ as well as $\mathbb{{{R}}}^{{{n}} \times {{k}}} \cong (\mathbb{{{R}}}^{{{n}} \times {{k}}})^*$ via $V \mapsto {\rm{tr}}\big(V^{\top} (\cdot) \big)$. Indeed, Equation (4.4) follows by the local formula for the canonical symplectic form $\omega_0$ on $T^*U$, see e.g. [16, Sec. 6.2], applied to the gobal chart $(U, \mathrm{id}_U) = (U, X_{ij})$.

4.1.   Lagrangian $2$-Form

We now calculate the Lagrangian $2$-from $\omega_{\widetilde{L}} = ({\rm{F}} \widetilde{L})^* \omega_0$. To this end, we first determine the fiber derivative ${\rm{F}} \widetilde{L} \colon T U \to T^* U$ and its tangent map.

Lemma 4.4. For $(X, V) \in TU$ the fiber derivative ${\rm{F}} \widetilde{L} \colon T U \to T^* U$ of $\widetilde{L}$ is given by

Proof. Let $(X, V), (X, W) \in TU$. We have $\big({\rm{F}} \widetilde{L} (X, V) \big) (X, W) = \langle V, W \rangle_X^{D, E}$ by the Definition of $\widetilde{L}$ and (4.1). Using the definition of $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ and exploiting properties of the trace we obtain

as desired.

Lemma 4.5. The tangent map $T ({\rm{F}} \widetilde{L}) \colon T (T U) \to T (T^* U)$ is given by

for $(X, V, Y, Z) \in T (T U) \cong U \times (\mathbb{{{R}}}^{n \times k})^3$, where we identify $T (T^* U) \cong U \times (\mathbb{{{R}}}^{n \times k})^* \times \mathbb{{{R}}}^{n \times k} \times (\mathbb{{{R}}}^{n \times k})^*$.

Proof. Let $(X, V, Y, Z) \in T(TU)$. The smooth curve $\gamma \colon (-\epsilon, \epsilon) \ni t \mapsto (X + t Y, V + t Z) \in TU$, for $\epsilon > 0$ sufficiently small, fulfills $\gamma(0) = (X, V)$ with $\dot{\gamma}(0) = (Y, Z)$. Then

This yields the desired result.

Lemma 4.6. The Lagrangian $2$-form $\omega_{\widetilde{L}} = ({\rm{F}} \widetilde{L})^* \omega_0 \in \Gamma^{\infty}\big(\Lambda^2 T^* (T U) \big)$ is given by

with $(X, V) \in T U \cong U \times \mathbb{{{R}}}^{{{n}} \times {{k}}}$ and $(X, V, Y, Z), (X, V, \widetilde{Y}, \widetilde{Z}) \in T_{(X, V)} T U$.

Proof. Using the formula for $\omega_0 \in \Gamma^{\infty}\big(\Lambda^2 T^* (T^* U)\big)$ from Remark 4.3, a straightforward calculation shows that $\omega_{\widetilde{L}} = ({\rm{F}} \widetilde{L})^* \omega_0$ 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 $({\rm{F}} \widetilde{L})^* \omega_0$.

4.2.   Sprays on $TU$

Next the spray $\widetilde{S} \in \Gamma^{\infty}\big(T (TU) \big)$ associated with $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ is calculated exploiting $\widetilde{S} = X_{E_{\widetilde{L}}}$, where $X_{E_{\widetilde{L}}}$ is the Lagrangian vector field. A closed form expression for $\widetilde{S}(X, V)$ is obtained for all $(X, V) \in \mathrm{St}_{{{n}}, {{k}}} \times \mathbb{{{R}}}^{{{n}} \times {{k}}} \subseteq T U$.

Lemma 4.7. For $(X, V) \in T U$ and $(X, V, Y, Z) \in T_{(X, V)} TU$ one has

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

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 $D, E \in \mathbb{{{R}}}^{k \times k}_{\mathrm{diag}}$ such that $D$ and $D + E$ are both invertible and let $W \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Moreover, let $U \subseteq \mathbb{{{R}}}^{{{n}} \times {{k}}}$ be open with the property from Lemma 3.8. Then for $X \in U$ the linear equation

has a unique solution in terms of $\widetilde{\Gamma}$. Moreover, for $X \in \mathrm{St}_{{{n}}, {{k}}}$, it is explicitly given by

Proof. For each $X \in U$ the linear map $\phi \colon \mathbb{{{R}}}^{{{n}} \times {{k}}} \ni \widetilde{\Gamma} \mapsto \widetilde{\Gamma} D + X X^{\top} \widetilde{\Gamma} E \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ is an isomorphism since the bilinear form

is non-degenerated by assumption. Hence (4.8) admits a unique solution. Now assume $X \in \mathrm{St}_{{{n}}, {{k}}}$. We briefly explain how (4.9) can be derived. By exploiting $X^{\top} X = I_k$, Equation (4.8) implies

Plugging $X^{\top} \widetilde{\Gamma} = X^{\top} W (D + E)^{-1}$ into (4.8) yields

A straightforward calculation shows that $\widetilde{\Gamma}$ is indeed a solution of (4.8).

Proposition 4.9. The spray $\widetilde{S} \in \Gamma^{\infty}\big(T (TU) \big)$ associated with the metric $\langle \cdot, \cdot \rangle_{(\cdot)}^{D, E}$ is given by

for all $(X, V) \in T U \cong U \times \mathbb{{{R}}}^{{{n}} \times {{k}}}$. Here $\widetilde{\Gamma} = \widetilde{\Gamma}_X(V, V) \in \mathbb{{{R}}}^{{{n}} \times {{k}}}$ depending on $(X, V) \in TU$ is the unique solution of the linear equation

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

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

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

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

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

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

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

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

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.

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

where

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

where the last equality holds due to

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

4.3.   Sprays on Stiefel Manifolds

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where the identity

is used. This yields

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

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

where the last equality follows due to

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

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.

5.   Pseudo-Riemannian gradients and pseudo-Riemannian Hessians

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.

5.1.   Pseudo-Riemannian Gradients

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

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

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

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

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

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

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

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

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

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

where the identity

is used to obtain the last equality.

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

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

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

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

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

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

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.

5.2.   Pseudo-Riemannian Hessians

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

6.   Second fundamental form and Levi-Civita covariant derivative

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.

6.1.   Levi-Civita covariant derivative on $U$

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

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

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

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

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

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

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

6.2.   Second fundamental form

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

Here we exploited

as well as

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

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 $\pi^{2 I_k + \nu I_k} = {\rm{skew}} \colon \mathbb{{{R}}}^{{{k}} \times {{k}}} \to \mathfrak{{{so}}}(k)$.

6.3.   Levi-Civita Covariant Derivative on $\mathrm{St}_{{{n}}, {{k}}}$

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 $X \in U$

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

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

Proof. Using Lemma 6.2 and Proposition 6.5 we compute

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

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

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

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 $-2 \neq \nu \in \mathbb{{{R}}}$

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

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

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

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)].

7.   Conclusions

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.

Acknowledgments

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

Conflict of interest

The author declares there is no conflict of interest.