A concircular vector field u on the unit sphere Sn+1 induces a vector field w on an orientable hypersurface M of the unit sphere Sn+1, simply called the induced vector field on the hypersurface M. Moreover, there are two smooth functions, f and σ, defined on the hypersurface M, where f is the restriction of the potential function ¯f of the concircural vector field u on the unit sphere Sn+1 to M and σ is defined as g(u,N), where N is the unit normal to the hypersurface. In this paper, we show that if function f on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field w has a certain lower bound, then a characterization of a small sphere in the unit sphere Sn+1 is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface M in the direction of the vector field w with a non-zero function σ.
Citation: Ibrahim Al-Dayel, Sharief Deshmukh, Olga Belova. Characterizing non-totally geodesic spheres in a unit sphere[J]. AIMS Mathematics, 2023, 8(9): 21359-21370. doi: 10.3934/math.20231088
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A concircular vector field u on the unit sphere Sn+1 induces a vector field w on an orientable hypersurface M of the unit sphere Sn+1, simply called the induced vector field on the hypersurface M. Moreover, there are two smooth functions, f and σ, defined on the hypersurface M, where f is the restriction of the potential function ¯f of the concircural vector field u on the unit sphere Sn+1 to M and σ is defined as g(u,N), where N is the unit normal to the hypersurface. In this paper, we show that if function f on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field w has a certain lower bound, then a characterization of a small sphere in the unit sphere Sn+1 is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface M in the direction of the vector field w with a non-zero function σ.
Research into understanding the geometry of hypersurfaces in the unit sphere Sn+1 is highly significant in differential geometry and has engaged the attention of several pioneering mathematicians [1,5,9,14,22,24,27,32,33,36]. It is worth noting there are still fascinating open problems in the geometry of hypersurfaces in the unit sphere, such as the Chern's problem on isometric hypersurfaces ([40], Problem 105). Over the period, several celebrated results in this area have been obtained; for example, Okumura [25] gave a criterion for a hypersurface of a unit sphere of constant mean curvature to be totally umbilical and Chen [7] characterized minimal hypersurfaces. In [2], the rigidity of compact-oriented hypersurfaces with constant scalar curvature isometrically immersed into the unit Euclidean sphere was studied. The papers [6,10] were devoted to the study of the Fisher–Marsden conjecture regarding the Kenmotsu manifold. In [3,11], the authors considered Ricci solitons. The Clifford hypersurface in a unit sphere was considered in [23,30]. A characterization of Euclidean spheres out of complete Riemannian manifolds was made by certain vector fields on complete Riemannian manifolds satisfying a partial differential equation on vector fields in [18]. Some characterizations of certain rank-one symmetric Riemannian manifolds by the existence of non-trivial solutions to certain partial differential equations on Riemannian manifolds are surveyed in [16].
There are two important hypersurfaces: the unit sphere Sn+1, namely the totally geodesic hypersurfaces Sn known as great spheres, and Sn(1α2), namely the small spheres. Some interesting results for the case of the unit sphere with constant curvature were received in [8,20,38,39]. Hypersurfaces were studied in [12,13,19,21,28,29,31,35,37,41]. In [4], authors have considered characterizing small spheres among compact hypersurfaces of the unit sphere Sn+1 using the Fischer–Marsden equation satisfied by the support function σ of the hypersurface.
It is well known that there are several concircular vector fields on the unit sphere Sn+1 obtained through tangential projections of constant vector fields on the ambient Euclidean space En+2. Such a concircular vector field u on Sn+1 satisfies ¯∇Xu=−¯fX, where X is a smooth vector field on Sn+1 and ¯f is a smooth function defined on Sn+1 called the potential function of the concircular vector field u. Given an orientable hypersurface M of the unit sphere Sn+1 with unit normal N and shape operator A, one can express the restriction of the concircular vector field u to M as u=w+σN, where w is tangent to the hypersurface M and σ=g(u,N) is a smooth function on M. We denote by f the restriction of the potential function ¯f to the hypersurface M. In this paper, we call the vector field w as the induced vector field on the hypersurface M, the function f as the associated function, and the function σ as the support function of the hypersurface. We show that the associated function f for the special hypersurface the small sphere Sn(c) satisfies the Fischer–Marsden equation.
Consider the unit sphere Sn+1 as the hypersurface of the Euclidean space Rn+2 with unit normal ξ and shape operator B=−I, where I denotes the identity operator. For the constant vector field Z=∂∂u1 on the Euclidean space Rn+2, where u1,...,un+2 are Euclidean coordinates on Rn+2, we denote the tangential projection of Z by u to the unit sphere Sn+1. Then, we have
Z=u+¯fξ, |
where ¯f=⟨Z,ξ⟩, ⟨,⟩ is the Euclidean metric on Rn+2. By differentiating the above equation with respect to a vector field X on the unit sphere Sn+1 and using the Gauss–Weingarten formulae for hypersurface, we have
¯∇Xu=−¯fX,grad¯f=u, |
where ¯∇ is the Riemannian connection on the unit sphere Sn+1 with respect to the canonical metric g and grad¯f is the gradient of the smooth function ¯f on Sn+1. The above equation shows that u is a concircular vector field on the unit sphere Sn+1.
Now, consider the small sphere (non-totally geodesic sphere) Sn(1α2) in the unit sphere Sn+1 defined by
Sn(1α2)={(u1,...,un+2):n+1∑i=1(ui)2=α2,un+2=√1−α2,0<α<1}. |
Then, it follows that Sn(1α2) is a hypersurface of the unit sphere Sn+1 with unit normal vector field ζ given by
ζ=(−√1−α2αu1,...,−√1−α2αun+1,α). |
We use the same letter g to denote the induced metric on the small sphere Sn(1α2) and denote the Riemannian connection with respect to the induced metric g by ∇. Then, by a simple computation, we have
¯∇Xζ=−√1−α2αX,X∈X(Sn(1α2)). | (2.1) |
That is, the shape operator A of the hypersurface Sn(1α2) is given by
A=√1−α2αI=HI, | (2.2) |
where H is the mean curvature of the hypersurface Sn(1α2). It is clear that H is a non-zero constant, as 0<α<1. Now, we utilize w to denote the tangential projection of the vector field u to the small sphere Sn(1α2) and define σ=g(u,ζ). Then, we have
u=w+σζ. | (2.3) |
However, using the definitions of u and ζ, we can easily see that
g(u,ζ)=−√1−α2αf, |
where f is the restriction of ¯f to Sn(1α2). Thus,
σ=−Hf. | (2.4) |
Differentiating Eq (2.3) and using the Gauss–Weingarten formulae for hypersurface, we conclude on using Eqs (2.1) and (2.2) and on equating tangential components, that
∇Xw=−(1+H2)fX,gradσ=−Hw, | (2.5) |
for X∈X(Sn(1α2)). Thus, in view of Eqs (2.4) and (2.5), the Laplace operator acting on the smooth function σ is given by
Δσ=−n(1+H2)σ. |
The Ricci tensor of the small sphere Sn(1α2) is given by
Ric=(n−1)(1+H2)g. |
Additionally, using Eqs (2.4) and (2.5), we have
gradf=w |
and consequently, we have
Hess(f)(X,Y)=−(1+H2)fg(X,Y),Δf=−n(1+H2)f. |
Thus, we see that for the function f, we have
(Δf)g+fRic=Hess(f). | (2.6) |
Thus, the function f satisfies the Fischer–Marsden equation [15,17,26,34].
Let M be an orientable hypersurface of the unit sphere Sn+1 with unit normal N and shape operator A. We denote the canonical metric on Sn+1 by g and the induced metric on M by the same letter g. Additionally, utilize ¯∇ and ∇ to denote the Riemannian connections on the unit sphere Sn+1 and the hypersurface M, respectively. Then, we have the following fundamental formulae for the hypersurface:
¯∇XY=∇XY+g(AX,Y)N,¯∇XN=−AX,X,Y∈X(M), | (3.1) |
where X(M) is the Lie-algebra of smooth vector fields on the hypersurface M. The curvature tensor R, the Ricci tensor Ric, and the scalar curvature of the hypersurface are given by
R(X,Y)Z=g(Y,Z)X−g(X,Z)Y+g(AY,Z)AX−g(AX,Z)AY,X,Y,Z∈X(M), |
Ric(X,Y)=(n−1)g(X,Y)+nHg(AX,Y)−g(AX,AY),X,Y∈X(M), | (3.2) |
τ=n(n−1)+n2H2−‖A‖2. | (3.3) |
The Codazzi equation for the hypersurface is
(∇A)(X,Y)=(∇A)(Y,X),X,Y∈X(M), |
where (∇A)(X,Y)=∇XAY−A∇XY. By using a local orthonormal frame {u1,..,un} on the hypersurface M and the mean curvature H=1ntrA in Eq (3.3), the following expression for the gradient of the mean curvature function H is given:
ngradH=n∑i=1(∇A)(ui,ui). | (3.4) |
Recall that on the unit sphere Sn+1, a concircular vector field u is defined using a constant vector field Z=∂∂u1 on the Euclidean space Rn+2 as Z=u+¯fξ, where the function ¯f=⟨Z,ξ⟩, ⟨,⟩ is the Euclidean metric on Rn+2 and we have
¯∇Xu=−¯fX,grad¯f=u,X∈X(Sn+1). |
We utilize f to denote the restriction of the function ¯f to the hypersurface M. We define a vector field w on the hypersurface M by
u=w+σN, | (3.5) |
that is, w is the tangential component of the concircular vector field u to the hypersurface M and the function σ=g(u,N). We call the vector field w the induced vector field on the hypersurface, the function σ as the support function of the hypersurface, and the function f as the associated function of the hypersurface. Taking covariant derivative in Eq (3.5), and using formulae in (3.1), we get
∇Xw=−fX+σAXandgradσ=−Aw,X∈X(M). | (3.6) |
Additionally, we have the tangential component [grad¯f]T=gradf and that the normal component [grad¯f]⊥=σN.
Theorem 1. Let M be an orientable, non-totally geodesic compact and connected hypersurface of the unit sphere Sn+1, n≥2, with mean curvature H, induced vector field w, and non-zero associated function f. Then, the potential function f is a non-trivial solution of the Fischer–Marsden Eq (2.6) and the inequality
∫M‖w‖2≥n∫M(1+H2)f2 |
holds if and only if H is a constant and M is isometric to the small sphere Sn(1+H2).
Proof. Suppose the associated function f of the hypersurface is a non-trivial solution of the Fischer–Marsden equation, that is,
(Δf)g+fRic=Hess(f). |
Taking trace in above equation, we conclude
Δf=−τn−1f. | (4.1) |
Now, using (3.5), we have gradf=w and Eq (3.6) implies divw=n(−f+σH). Thus, Δf=n(−f+σH) and combining it with (4.1), we have
−τn−1f=n(−f+σH). |
Using Eq (3.3) in above equation, we conclude
1n−1(‖A‖2−nH2)f2=nσfH+nH2f2. | (4.2) |
Note by on using gradf=w and divw=n(−f+σH), we have div(fw)=‖w‖2−nf2+nfσH. Thus, Eq (4.2) becomes
1n−1(‖A‖2−nH2)f2=nf2+nH2f2−‖w‖2+div(fw) |
and integrating above equation, we have
1n−1∫M(‖A‖2−nH2)f2=n∫M(1+H2)f2−∫M‖w‖2. |
Note that owing to Schwartz's inequality ‖A‖2≥nH2, the integral on the left hand side is non-negative, and consequently, using the condition in the statement, we conclude that
1n−1∫M(‖A‖2−nH2)f2=0. |
Thus, the Schwartz's inequality is actually equality ‖A‖2=nH2, which holds if and only if A=HI. We compute (∇A)(X,Y)=X(H)Y and summing the last equation over a local orthonormal frame {u1,..,un} on M, we conclude that
n∑i=1(∇A)(ui,ui)=gradH |
and combining this equation with Eq (3.4), we obtain ngradH=gradH. Since n≥2, we get gradH=0, that is, H is a constant. Hence, we see that M is isometric to the small sphere Sn(1+H2).
Conversely, suppose that the hypersurface M is isometric to the small sphere Sn(1+H2). Then, from the introduction, it follows that the associated function f satisfies the Fischer–Marsden equation (cf. (2.6)) and that Δf=−n(1+H2)f implies that f has to be a non-trivial solution, for otherwise, we shall have f=0 and w=0, which by equation (2.4) will imply σ=0, and in turn Eq (2.3) will imply u=0. It will imply that ¯f=0, and consequently, Z=0, a contradiction. Moreover, we have
fΔf=−n(1+H2)f2, |
which on integrating by parts, gives
∫M‖gradf‖2=n(1+H2)∫Mf2. |
Using w=gradf, in above equation gives the equality
∫M‖w‖2=n(1+H2)∫Mf2. |
Hence, the converse holds.
In the following result, we shall use a lower bound on the integral of the Ricci curvature Ric(w,w) of a compact non-totally geodesic hypersurface with non-zero potential function σ of the unit sphere Sn+1, to find a characterization of a small sphere. Indeed we prove:
Theorem 2. Let M be an orientable non-totally geodesic compact and connected hypersurface of the unit sphere Sn+1, n≥2, with mean curvature H, induced vector field w, non-zero support function σ. Then, the inequality
∫MRic(w,w)≥(n−1)∫M(n(σ2H2−f2)+2‖w‖2) |
holds if and only if H is a constant and M is isometric to the small sphere Sn(1+H2).
Proof. Suppose M is an orientable non-totally geodesic compact and connected hypersurface of the unit sphere Sn+1, n≥2, with a mean curvature H, induced vector field w, and non-zero support function σ with the inequality
∫MRic(w,w)≥(n−1)∫M(n(σ2H2−f2)+2‖w‖2) | (4.3) |
holds. Note that, by differentiating gradσ=−Aw, and using Eq (3.6), we have the following expression for the Hessian operator Aσ:
AσX=−∇XAv=−[(∇A)(X,w)+A(−fX+σAX)],X∈X(M), |
that is,
AσX=−(∇A)(X,w)+fAX−σA2X,X∈X(M). | (4.4) |
For a local orthonormal frame {u1,..,un} on M, using symmetry of the shape operator A and Eq (3.4), we have
n∑i=1g((∇A)(ui,w),ui)=n∑i=1g(w,(∇A)(ui,ui))=nw(H). |
Taking trace in Eq (4.4), while using above equation, we get the following expression for the Laplacian Δσ
Δσ=−nw(H)+nfH−σ‖A‖2, |
that is,
σΔσ=−nσw(H)+nσfH−σ2‖A‖2. | (4.5) |
Note that Eq (3.6) gives, divw=n(−f+σH), which implies
divH(σw)=σw(H)+Hdiv(σw)=σw(H)+H(w(σ)+nσ(−f+σH)), |
which on using gradσ=−Aw, gives
divH(σw)=σw(H)−Hg(Aw,w)−nHσf+nσ2H2. |
Inserting the value of σw(H) from above equation in Eq (4.5), we get
σΔσ=−n(divH(σw)+Hg(Aw,w)+nHσf−nσ2H2)+nσfH−σ2‖A‖2. |
Integrating by parts the above equation, we get
−∫M‖gradσ‖2=∫M(−nHg(Aw,w)−n(n−1)σfH+n2σ2H2−σ2‖A‖2). | (4.6) |
Now, using Eq (3.2) and gradσ=−Aw, that is,
−∫M‖gradσ‖2=∫M(Ric(w,w)−(n−1)‖w‖2−nHg(Aw,w)) |
in Eq (4.6), we get
∫M(Ric(w,w)−(n−1)‖w‖2)=∫M(−n(n−1)σfH+n2σ2H2−σ2‖A‖2), |
that is,
∫Mσ2(‖A‖2−nH2)=∫M(n(n−1)σ2H2−n(n−1)σfH+(n−1)‖w‖2−Ric(w,w)). | (4.7) |
Also, using gradf=w, we get div(fw)=‖w‖2+fdiv(w)=‖w‖2+nf(−f+σH), that is,
nfσH=div(fw)+nf2−‖w‖2. |
Inserting above equation in the Eq (4.7), we get
∫Mσ2(‖A‖2−nH2)=∫M(n(n−1)(σ2H2−f2)+2(n−1)‖w‖2−Ric(w,w)), |
that is,
∫Mσ2(‖A‖2−nH2)=∫M((n−1)[n(σ2H2−f2)+2‖w‖2]−Ric(w,w)). |
Using inequality (4.3), we conclude
∫Mσ2‖A−HI‖2≤0, |
that is, σ2‖A−HI‖2=0, which together with σ≠0 implies A=HI. Then, as n≥2, and the argument given in the Proof of above Theorem, we get H is constant and M is isometric to Sn(1+H2).
Conversely, as M is non-totally geodesic hypersurface isometric to Sn(1+H2), by Eq (2.4), we see σ≠0. Also, we have
Ric(w,w)=(n−1)(1+H2)‖w‖2 | (4.8) |
and Eq (2.5) implies
divw=−n(1+H2)f. |
By using div(fw)=w(f)+fdivw=‖w‖2−n(1+H2)f2, we get
∫M‖w‖2=n(1+H2)∫Mf2. | (4.9) |
Using Eq (4.9) in the integral of Eq (4.8), we have
∫MRic(w,w)=n(n−1)(1+H2)2∫Mf2. | (4.10) |
Now, using Eqs (2.4) and (4.9), we get
(n−1)∫M(n(σ2H2−f2)+2‖w‖2)=(n−1)∫M(n(f2H4−f2)+2n(1+H2)f2), |
that is,
(n−1)∫M(n(σ2H2−f2)+2‖w‖2)=n(n−1)(1+H2)2∫Mf2. | (4.11) |
Equations (4.10) and (4.11) imply
∫MRic(w,w)=(n−1)∫M(n(σ2H2−f2)+2‖w‖2). |
Hence, all the requirements of the statement hold.
In this paper, we asked whether the Fischer–Marsden equation is satisfied by the associated function f could be used to characterize small spheres in the unit sphere Sn+1.
In the first result of this paper, we answered this question and obtained a characterization for a small sphere.
In yet other result, we obtained an interesting characterization of the small sphere using an appropriate lower bound on the integral of the Ricci curvature Ric(w,w).
It is known that for the small sphere Sn(1+H2) in the unit sphere Sn+1, its support function σ and the associated function f satisfies (see Eq (2.4))
σ=−Hf. |
This initiates a natural question: Does a non-totally geodesic compact hypersurface M with support function σ, associated function f and mean curvature H of the unit sphere Sn+1 satisfying the equation σ=−Hf necessarily isometric to the small sphere Sn(1+H2)? Answering this question will be an interesting future study in the geometry of hypersurfaces of the unit sphere Sn+1.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program No. RP-21-09-10.
The authors declare no conflicts of interest.
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