In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field ω on a compact and connected hypersurface N of the Euclidean space Rm+1 with a mean curvature α constant along the integral curves of ω and a shape operator T satisfying T(ω)=αω implies that α is a constant and N is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field v on a compact and connected hypersurface N of a Euclidean space Rm+1 gives a nonzero function σ=g(Tv,v) with shape operator T, and the integral of the function mασRic(v,v) has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface N with support ρ and basic vector field u, the integral of the Ricci curvature Ric(u,u) has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.
Citation: Hanan Alohali, Sharief Deshmukh. Some generic hypersurfaces in a Euclidean space[J]. AIMS Mathematics, 2024, 9(6): 15008-15023. doi: 10.3934/math.2024727
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In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field ω on a compact and connected hypersurface N of the Euclidean space Rm+1 with a mean curvature α constant along the integral curves of ω and a shape operator T satisfying T(ω)=αω implies that α is a constant and N is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field v on a compact and connected hypersurface N of a Euclidean space Rm+1 gives a nonzero function σ=g(Tv,v) with shape operator T, and the integral of the function mασRic(v,v) has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface N with support ρ and basic vector field u, the integral of the Ricci curvature Ric(u,u) has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.
The geometry of hypersurfaces lies at the foundation of differential geometry, it started with the theory of curves and surfaces in the Euclidean 3-space R3 [11]. Given an orientable immersed hypersurface N in the Euclidean space Rm+1 with immersion φ:N→Rm+1, we have the unit normal ζ, the shape operator T, the support ρ=⟨φ,ζ⟩ a smooth function defined on the hypersurface N and the mean curvature α, given by mα=trT being trace of the shape operator T [11]. If the hypersurface N of the Euclidean space Rm+1 is compact, then we have the following well-known Minkowski's formula:
∫N(1+ρα)=0. | (1.1) |
As an outcome of Minkowski's formula, we conclude that there are no compact minimal hypersurfaces (hypersurfaces with mean curvature α=0) in the Euclidean space Rm+1.
Among compact hypersurfaces of Euclidean spaces, important are the Euclidean spheres Sm(c) of constant curvature c, with the imbedding φ:Sm(c)→Rm+1, φ(x)=x, shape operator T=−√cI, and unit normal ζ=√cφ. Taking a as a nonzero constant vector field on Rm+1, we can express it as a=u+fζ, where f=⟨a,ζ⟩ and u is the tangential projection of a on the sphere Sm(c). Letting g be the induced metric and ∇ the Riemannian connection on the sphere Sm(c) and differentiating the equation a=u+fζ with respect to the vector field E on Sm(c), we have
∇Eu=−√cfE, ∇f=√cu, | (1.2) |
where ∇f is the gradient of f.
On an odd dimensional sphere S2m−1(c) with imbedding φ:S2m−1(c)→R2m with unit normal ζ=√cφ, shape operator T=−√cI, apart from the above vector field u, there is a unit vector field v defined on S2m−1(c) by
v=Jζ, | (1.3) |
where J is the complex structure on the Euclidean space R2m. Differentiating the above equation using the Euclidean connection D with respect to a vector field E on S2m−1(c), one confirms
∇Ev−√c⟨E,v⟩ζ=√cJE, |
that is,
∇Ev=√c(JE)T, | (1.4) |
where (JE)T is the tangential projection of JE on S2m−1(c).
Given an immersed hypersurface N of the Euclidean space Rm+1, the natural tools for studying the geometry of N are the shape operator T, the mean curvature α, the curvature tensor R, the Ricci tensor Ric, the Ricci operator S, and the scalar curvature τ of N. In [8], it is shown that a compact hypersurface M of the Euclidean space Rm+1 satisfies the inequality
‖T‖2τ≥12‖R‖2+‖S‖2+2m(m−1)‖∇α‖2, |
if and only if α is a constant and N is isometric to the n -sphere Sm(α2). Also, in [9], the position vector field φ of a compactly immersed hypersurface N in the Euclidean space Rm+1 with immersion φ:N→Rm+1 and unit normal ζ was used to define a vector field u on the hypersurface N as the tangential projection of the position vector field φ that leads to the integral formula
∫N{Ric(u,u)+m(m−1)−ρ2τ}=0, |
where ρ=⟨φ,ζ⟩ is the support of N. In [7,9], the above integral was used, which led to many important geometric implications on the compact hypersurface N of the Euclidean space Rm+1. Moreover, in [8], it is shown that a compact hypersurface N of positive Ricci curvature in the Euclidean space Rm+1 with scalar curvature τ≤λ1(m−1) is necessarily isometric to the sphere Sm(c), where λ1 is the first nonzero eigenvalue of the Laplace operator Δ of N with respect to the induced metric.
Recently, there has been a trend toward studying the geometry of the hypersurfaces in Rm+1, as the graphs of the smooth functions h:Rm+1→R are called the translation hypersurfaces. The focus, in translation hypersurface N of the Euclidean space Rm+1, is on the property function h:Rm+1→R, whose graph is N. In [18], translation hypersurfaces of Rm+1 are studied, whose Gauss-Kronecker curvature depends on either its first p variables or on the rest q variables, where m=p+q, and conditions on a translation hypersurface to have Gauss-Kronecker zero curvature are found. If a translation hypersurface N is defined as the graph of the function h:Rm+1→R with h satisfying certain additional conditions, then it is called a separable hypersurface. Separable hypersurfaces in the Euclidean space Rm+1 have an interesting geometry, as studied in [6,12,13,19]. A complete classification of separable hypersurfaces with zero Gauss-Kronecker curvature in the Euclidean space Rm+1 is obtained in [6].
In this paper, we are interested in studying the impact of the existence of a concircular vector field as well as a Killing vector field on the immersed hypersurface N of the Euclidean space Rm+1. A vector field ω on a Riemannian manifold (N,g) is a concircular vector field if
∇Eω=σE, E∈Ψ(N), |
where σ is a function on N and Ψ(N) is the space of smooth vector fields on N. We shall use the abbreviation CLVF for a concircular vector field. It is known that a CLVF ω on a Riemannian manifold (N,g) influences the geometry of (N,g) [4,5]. Moreover, a CLVF ω has a role in general relativity [3].To understand the role of CLVF in relativity, recall that m -dimensional generalized Robertson-Walker space-time, m>3, is the warped product I×h2M, with Lorentz metric g=−dt2+h2g∗, where I is an interval h:I→R is a positive smooth function and (M,g∗) is a Riemannian manifold with dimM=(m−1). In [3], Chen has proved a very significant result involving a CLVF, namely: A Lorentzian manifold admits a nontrivial timelike CLVF if and only if it is a generalized Robertson-Walker space-time. Note that Eq (1.2) shows that the vector field u is a CLVF on the sphere Sm(c) with potential function σ=−√cf and naturally the shape operator T of the sphere Sm(c) as a hypersurface of the Euclidean space Rm+1 satisfies T(u)=αu, where α=−√c is the mean curvature of Sm(c). This naturally raises a question: Is a compact and connected hypersurface N with shape operator T and mean curvature α of the Euclidean space Rm+1 admitting a nonzero CLVF u satisfying T(u)=αu, u(α)=0, necessarily isometric to Sm(c)? In Section 3, we show that this question has an affirmative answer, and indeed, we show that the converse is also true.
Similarly, a vector field ω on an m-dimensional Riemannian manifold (N,g) is said to be a Killing vector field if
\begin{equation*} \mathtt{£} _{{\omega }}g = 0\text{, } \end{equation*} |
and we shall use the abbreviation KGVF for a Killing vector field. Note that the presence of a KGVF {\omega } on (N, g) influences its geometry as well as topology [2,14,17,21]. Note that the unit vector field \mathbf{v} on the sphere S^{2m-1}(c) satisfies Eq (1.4), which leads to
\begin{equation*} \mathtt{£} _{\mathbf{v}}g = 0\text{, } \end{equation*} |
that is, \mathbf{v} is a unit KGVFon the sphere S^{2m-1}(c) . We see that \sigma = g\left(T\mathbf{v}, \mathbf{v}\right) = -\sqrt{c} is a constant, and the following holds:
\begin{equation} \int_{S^{2m-1}(c)}m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v} \right) = \int_{S^{2m-1}(c)}\left( m(m-1)\alpha ^{2}\sigma ^{2}-\left\Vert \nabla \sigma \right\Vert ^{2}\right) . \end{equation} | (1.7) |
This raises the next question: Does a compact and connected hypersurface N with shape operator T , mean curvature \alpha , induced metric g , admitting a unit KGVF \mathbf{v} , of a Euclidean space R^{m+1} with nonzero function \sigma = g\left(T\mathbf{v}, \mathbf{v}\right) satisfying Eq (1.7) necessarily imply m is odd, \alpha a constant, and M isometric to S^{2m-1}(c) ? In Section 4 of this paper, we answer this question and find a characterization of the sphere S^{2m-1}(c) .
Finally, in the last section, we consider an immersed compact and connected hypersurface N in the Euclidean space R^{m+1} with immersion \varphi :N\rightarrow R^{m+1} , unit normal \zeta , and shape operator T . Then, we express the position vector field \varphi as \varphi = \mathbf{u} +f\zeta , where f = \langle \varphi, \zeta \rangle is the support function of the hypersurface. In the last section, we shall prove that for a compact and connected hypersurface N with nonzero support function and if the following condition holds
\begin{equation*} \int_{N}Ric\left( \mathbf{u}, \mathbf{u}\right) \geq \frac{m-1}{m} \int_{N}\left( {div}\mathbf{u}\right) ^{2}\text{, } \end{equation*} |
then the mean curvature \alpha is a constant and N is the sphere S^{m}\left(\alpha ^{2}\right) .
Let N be an orientable hypersurface of the Euclidean space R^{m+1} with unit normal \zeta , shape operator T . We denote the Euclidean metric by \langle, \rangle and by g the induced metric on N , and by \nabla and D , the Riemannian connection with respect to g and the Euclidean connection, respectively. Then, we have [11]
\begin{equation} D_{E}F = \nabla _{E}F+g\left( TE, F\right) \zeta \text{, }\;\;D_{E}\zeta = -TE \text{, }\;\;E, F\in \Psi \left( N\right) \text{, } \end{equation} | (2.1) |
where \Psi \left(N\right) is the space of smooth vector fields on N . The curvature tensor field of the hypersurface N is given by
\begin{equation} R(E, F)G = g\left( TF, G\right) TE-g\left( TE, G\right) TF\text{, }\;\;E, F, G\in \Psi \left( N\right) \text{, } \end{equation} | (2.2) |
and the Ricci tensor of N has the expression
\begin{equation} Ric\left( E, F\right) = m\alpha g\left( TE, F\right) -g\left( TE, TF\right) \text{, } \end{equation} | (2.3) |
where \alpha is the mean curvature of the hypersurface N , given by m\alpha = trT , the trace of the shape operator T . For a local orthonormal frame \left\{ w_{k}\right\} _{1}^{m} on the hypersurface, the scalar curvature \tau of the hypersurface N is given by
\begin{equation*} \tau = \sum\limits_{k = 1}^{m}Ric\left( w_{k}, w_{k}\right) \text{, } \end{equation*} |
and combining the above equation with (2.3), gives
\begin{equation} \tau = m^{2}\alpha ^{2}-\left\Vert T\right\Vert ^{2}\text{, } \end{equation} | (2.4) |
where
\begin{equation*} \left\Vert T\right\Vert ^{2} = \sum\limits_{k = 1}^{m}g\left( Tw_{k}, Tw_{k}\right) . \end{equation*} |
The Codazzi equation of the hypersurface N is given by
\begin{equation} \left( \nabla _{E}T\right) F = \left( \nabla _{F}T\right) E\text{, } \;\;\;E, F\in \Psi \left( N\right) \text{, } \end{equation} | (2.5) |
where \left(\nabla _{E}T\right) F = \nabla _{E}TF-T\left(\nabla _{E}F\right) . Note that, as the shape operator T is symmetric, we have for E\in \Psi (N) and a local frame \left\{ w_{k}\right\} _{1}^{m} ,
\begin{align*} mE\left( \alpha \right) & = \sum\limits_{k = 1}^{m}Eg\left( Tw_{k}, w_{k}\right) = \sum\limits_{k = 1}^{m}g\left( \left( \nabla _{E}T\right) \left( w_{k}\right) , w_{k}\right) +2\sum\limits_{k = 1}^{m}g\left( Tw_{k}, \nabla _{E}w_{k}\right) \notag \\ & = \sum\limits_{k = 1}^{m}g\left( \left( \nabla _{w_{k}}T\right) \left( E\right) , w_{k}\right) +2\sum\limits_{k = 1}^{m}g\left( Tw_{k}, \nabla _{E}w_{k}\right) \notag \\ & = \sum\limits_{k = 1}^{m}g\left( E, \left( \nabla _{w_{k}}T\right) \left( w_{k}\right) \right) +2\sum\limits_{k = 1}^{m}g\left( Tw_{k}, \nabla _{E}w_{k}\right) , \end{align*} | (2.6) |
and using the facts that
\begin{equation*} Tw_{k} = \sum\limits_{j = 1}^{m}\lambda _{k}^{j}w_{j}\text{, }\;\;\nabla _{E}w_{k} = \sum\limits_{i = 1}^{m}\omega _{k}^{i}(E)w_{i}\text{, } \end{equation*} |
where \left(\lambda _{k}^{j}\right) is a symmetric matrix and \omega _{k}^{i} are connection forms, which are skew symmetric, that is, \omega _{k}^{i}+\omega _{i}^{k} = 0 ; in Eq (2.6), we conclude
\begin{equation*} mE\left( \alpha \right) = \sum\limits_{k = 1}^{m}g\left( E, \left( \nabla _{w_{k}}T\right) \left( w_{k}\right) \right) . \end{equation*} |
Therefore, the gradient of \alpha has the expression
\begin{equation} \nabla \alpha = \frac{1}{m}\sum\limits_{k = 1}^{m}\left( \nabla _{w_{k}}T\right) \left( w_{k}\right) . \end{equation} | (2.7) |
Let {\omega } be a CLVF on an m -dimensional Riemannian manifold (N, g) . Then, we have
\begin{equation} \nabla _{E}{\omega } = \sigma E\text{, }\;\;\;E\in \Psi \left( N\right) \text{, } \end{equation} | (2.8) |
where \sigma is the potential function of the CLVF { \omega } . A CLVF {\omega } on (N, g) is said to be nontrivial if it is not parallel. We have the following expression for the curvature tensor field of (N, g) involving the CLVF { \omega }
\begin{equation*} R\left( E, F\right) {\omega } = E\left( \sigma \right) F-F\left( \sigma \right) E\text{, }\;\;\;E, F\in \Psi \left( N\right) . \end{equation*} |
Taking the trace in the above equation, we see that the Ricci tensor of \left(N, g\right) is given by
\begin{equation} Ric\left( E, {\omega }\right) = -(m-1)E\left( \sigma \right) \text{ , }\;\;\;E\in \Psi \left( N\right) . \end{equation} | (2.9) |
The Ricci operator S of the Riemannian manifold \left(N, g\right) is given by
\begin{equation*} Ric\left( E, F\right) = g\left( SE, F\right) , \end{equation*} |
and thus, using Eq (2.9), we see that the Ricci operator S operating on the CLVF {\omega } is given by
\begin{equation} S\left( {\omega }\right) = -(m-1)\nabla \sigma \text{, } \end{equation} | (2.10) |
where \nabla \sigma is the gradient of \sigma .
Now, consider a KGVF \mathbf{v} on an m -dimensional Riemannian manifold (N, g) that satisfies [11]
\begin{equation} \mathtt{£} _{\mathbf{v}}g = 0. \end{equation} | (2.11) |
Note that the flow of a KGVF on a Riemannian manifold consists of isometries, and therefore, its presence influences both the topology and geometry of the manifold on which they live. For instance, if \mathbf{v} is a KGVFon a Riemannian manifold (N, g) , then the scalar curvature \tau of (N, g) is constant along the integral curves of \mathbf{v} . It is known that, if a positively curved Riemannian manifold \left(N, g\right) admits a nontrivial KGVF, then its fundamental group contains a cyclic subgroup of constant index depending on \dim N [17]. Also, the presence of a nontrivial KGVFinfluences the dimension of the Riemannian manifold on which they live. For instance, on the even-dimensional unit sphere S^{2m} there does not exist a unit KGVF, where as on S^{2m+1} a unit KGVF exists [2,11]. Moreover, the presence of a nontrivial KGVFon a compact Riemannian manifold (N, g) does not allow it to have a non-positive Ricci curvature [11].
There is a skew-symmetric operator \phi associated with the KGVF \mathbf{v} on (N, g) that satisfies
\begin{equation} \nabla _{E}\mathbf{v} = \phi E\text{, }\;\;\;E\in \Psi \left( N\right) , \end{equation} | (2.12) |
and that the covariant derivative of the operator \phi is given by
\begin{equation} \left( \nabla _{E}\phi \right) \left( F\right) = R\left( E, \mathbf{v}\right) F \text{, }\;\;\;E, F\in \Psi \left( N\right) . \end{equation} | (2.13) |
It is clear from Eq (2.12) that \mathbf{v} , being a unit KGVF on (N, g) , satisfies
\begin{equation} \phi \mathbf{v} = 0. \end{equation} | (2.14) |
Note that the flow of a KGVF \mathbf{v} on an m -dimensional Riemannian manifold (N, g) consists of isometries of (N, g) . Now suppose that N is an orientable hypersurface of the Euclidean space R^{m+1} with shape operator T , mean curvature \alpha , and induced metric. Suppose that there is a unit KGVF \mathbf{v} on the hypersurface N . We say that the shape operator T of the hypersurface is invariant under the unit KGVF \mathbf{v} if
\begin{equation} \psi _{t}^{\ast }\left( T\right) = T\circ d\psi _{t}\text{, } \end{equation} | (2.15) |
where \left\{ \psi _{t}\right\} is the flow of the unit KGVF \mathbf{v} .
Lemma 1. Let \mathbf{v} be a unit KGVF on the hypersurface N of the Euclidean space R^{m+1} such that the shape operator T is invariant under \mathbf{v} . Then the shape operator satisfies
\begin{equation*} \left( \nabla _{E}T\right) \left( \mathbf{v}\right) = \phi \left( TE\right) -T\left( \phi E\right) \mathit{\text{, }}\;\;\;E\in \Psi \left( N\right) \mathit{.} \end{equation*} |
Proof. Since T is invariant under \mathbf{v} , Eq (2.15) implies
\begin{equation*} \mathtt{£} _{\mathbf{v}}T = 0\text{, } \end{equation*} |
which gives
\begin{equation*} \lbrack \mathbf{v}, TE] = T[\mathbf{v}, E]\text{, }\;\;\;E\in \Psi \left( N\right) \text{, } \end{equation*} |
that is, in view of Eq (2.12), we have
\begin{equation*} \left( \nabla _{\mathbf{v}}T\right) \left( E\right) = \phi \left( TE\right) -T\left( \phi E\right) \text{, }\;\;\;E\in \Psi \left( N\right) . \end{equation*} |
Combining the above equation with Eq (2.5), we get the result.
In this section, we are interested in studying the impact of a nonzero CRVF \mathbf{\omega } with potential \sigma on a compact hypersurface N of the Euclidean space R^{m+1} . We would like to recall that given a smooth curve \beta :I\rightarrow N on the hypersurface N with mean curvature \alpha , we get a smooth function f:I\rightarrow R defined by f = \alpha \circ \beta and if f is a constant function, we say the mean curvature \alpha is a constant along the curve \beta on the hypersurface. Naturally, if the mean curvature \alpha is a constant, then it will be constant along each curve on the hypersurface. However, mean curvature \alpha being constant along some curves on hypersurface N does not imply that \alpha is a constant on N . In the following result, we shall assume that the mean curvature \alpha is a constant along the integral curves of the CRVF \mathbf{\omega } , which is a weaker condition than asking if the mean curvature \alpha is a constant. Indeed, we prove the following:
Theorem 1. A compact and connected hypersurface N of the Euclidean space R^{m+1} , m > 1 , admits a nonzero nontrivial CRVF {\omega } such that the mean curvature \alpha is constant along the integral curves of {\omega } and the shape operator T satisfies T\left({\omega }\right) = \alpha {\omega } , if and only if \alpha is a constant and N is isometric to S^{m}\left(\alpha ^{2}\right) .
Proof. Suppose that the compact and connected hypersurface N of R^{m+1} , m > 1 , admits a nonzero nontrivial CRVF {\omega } with potential \sigma , such that the mean curvature \alpha is constant along the integral curves of {\omega } and the shape operator T satisfies
\begin{equation} T\left( {\omega }\right) = \alpha {\omega }. \end{equation} | (3.1) |
Then we have
\begin{equation} {\omega }\left( \alpha \right) = 0. \end{equation} | (3.2) |
Using Eqs (2.8) and (3.1), we get
\begin{equation*} \left( \nabla _{E}T\right) \left( {\omega }\right) = E\left( \alpha \right) {\omega }+\sigma \alpha E-\sigma TE\text{, }\;\;\;E\in \Psi \left( N\right) \text{, } \end{equation*} |
that is,
\begin{equation} \sigma \left( TE-\alpha E\right) = E\left( \alpha \right) {\omega } -\left( \nabla _{E}T\right) \left( {\omega }\right) \text{, } \;\;\;E\in \Psi \left( N\right) . \end{equation} | (3.3) |
Now, using a local frame \left\{ w_{k}\right\} _{1}^{m} on the hypersurface N , we have
\begin{equation*} \sigma ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = \sum\limits_{k = 1}^{m}g\left( \sigma \left( Tw_{k}-\alpha w_{k}\right) , \sigma \left( Tw_{k}-\alpha w_{k}\right) \right), \end{equation*} |
and employing Eq (3.3) in the above equation leads to
\begin{align*} \sigma ^{2}\left\Vert T-\alpha I\right\Vert ^{2}& = \sum\limits_{k = 1}^{m}g\left( w_{k}\left( \alpha \right) {\omega } -\left( \nabla _{w_{k}}T\right) \left( {\omega }\right) , w_{k}\left( \alpha \right) {\omega }-\left( \nabla _{w_{k}}T\right) \left( {\omega }\right) \right) \notag \\ & = \left\Vert \nabla \alpha \right\Vert ^{2}\left\Vert {\omega } \right\Vert ^{2}+\sum\limits_{k = 1}^{m}g\left( \left( \nabla _{w_{k}}T\right) \left( {\omega }\right) , \left( \nabla _{w_{k}}T\right) \left( {\omega }\right) \right) -2g\left( \nabla \alpha , \left( \nabla _{{\omega }}T\right) \left( {\omega } \right) \right) . \end{align*} | (3.4) |
Moreover, Eqs (3.1) and (3.2) give
\begin{equation} \left( \nabla _{{\omega }}T\right) \left( {\omega }\right) = \nabla _{{\omega }}\left( \alpha {\omega }\right) -T\left( \sigma {\omega }\right) = 0. \end{equation} | (3.5) |
Next, using Eq (3.1), we compute
\begin{equation*} \left( \nabla _{w_{k}}T\right) \left( {\omega }\right) = w_{k}\left( \alpha \right) {\omega }+\alpha \sigma w_{k}-\sigma T\left( w_{k}\right) \text{, } \end{equation*} |
which, on using Eq (3.2), on some simplifications, gives
\begin{equation} \sum\limits_{k = 1}^{m}g\left( \left( \nabla _{w_{k}}T\right) \left( { \omega }\right) , \left( \nabla _{w_{k}}T\right) \left( {\omega } \right) \right) = \left\Vert \nabla \alpha \right\Vert ^{2}\left\Vert { \omega }\right\Vert ^{2}+\sigma ^{2}\left\Vert T\right\Vert ^{2}-m\sigma ^{2}\alpha ^{2}. \end{equation} | (3.6) |
Thus, Eqs (3.4)–(3.6), yield
\begin{equation} \sigma ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = 2\left\Vert \nabla \alpha \right\Vert ^{2}\left\Vert {\omega }\right\Vert ^{2}+\sigma ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) . \end{equation} | (3.7) |
Also, we have
\begin{eqnarray*} \left\Vert T-\alpha I\right\Vert ^{2} & = &\sum\limits_{k = 1}^{m}g\left( \left( Tw_{k}-\alpha w_{k}\right) , \left( Tw_{k}-\alpha w_{k}\right) \right) \\ & = &\left\Vert T\right\Vert ^{2}+m\alpha ^{2}-2\alpha \sum\limits_{k = 1}^{m}g\left( Tw_{k}, w_{k}\right) \\ & = &\left\Vert T\right\Vert ^{2}-m\alpha ^{2}. \end{eqnarray*} |
Substituting this last equation in Eq (3.7), we arrive at
\begin{equation*} 2\left\Vert \nabla \alpha \right\Vert ^{2}\left\Vert {\omega } \right\Vert ^{2} = 0, \end{equation*} |
and as {\omega } is a nonzero vector field on the connected hypersurface N , we conclude that \alpha is a constant. Now, using Eq (3.1) in the expression of the Ricci operator S of the hypersurface N , we get
\begin{equation*} S\left( {\omega }\right) = m\alpha T\left( {\omega }\right) -T^{2}\left( {\omega }\right) = (m-1)\alpha ^{2}{\omega }\text{. } \end{equation*} |
Combining this equation with Eq (2.10), we have
\begin{equation*} \nabla \sigma = -\alpha ^{2}{\omega }. \end{equation*} |
Differentiating the above equation with respect to a vector field E on N , and using Eq (2.8), we get
\begin{equation} \nabla _{E}\nabla \sigma = -\alpha ^{2}\sigma E\text{, }\;\;\;E\in \Psi \left( N\right) . \end{equation} | (3.8) |
The mean curvature \alpha is a constant; it has to be a nonzero constant as N is a compact hypersurface by virtue of the fact that there are no compact minimal hypersurfaces in the Euclidean space R^{m+1} , which is guaranteed by Minkowski's formula (1.1). Now, it remains to show that the potential \sigma cannot be a constant. To achieve it, we see that Eq (2.8) implies {div}{ \omega } = m\sigma , which, on integration, yields
\begin{equation*} \int_{N}\sigma = 0, \end{equation*} |
and if \sigma were a constant, it should give \sigma = 0 , which would make \omega a trivial CRVF, which is a contradiction. Hence, \sigma is a non-constant function. Hence, Eq (3.8) is Obata's differential equation [15,16], which confirms that N is isometric to S^{n}\left(\alpha ^{2}\right) .
Conversely, suppose N is isometric to S^{n}\left(c\right) . Then, by Eq (1.2), there is a CRVF \mathbf{u} on S^{m}(c) with potential \sigma = -\sqrt{c}f . We claim that \mathbf{u} is a nonzero and nontrivial CRVF \mathbf{\ } on S^{m}(c) . If \mathbf{u} = 0 , then by Eq (1.2), it will follow that f = 0 , and consequently, the constant vector \mathbf{a} = 0 , which is contrary to our assumption that \mathbf{a} is a nonzero constant vector field on the Euclidean space R^{m+1} . Similarly, if \mathbf{u} is parallel, then by Eq (1.2), we have f = 0 , and the second equation in Eq (1.2) will imply \mathbf{u} = 0 , which is a contradiction. Hence, \mathbf{u} is a nonzero and nontrivial CRVF \mathbf{\ } on S^{m}(c) , which satisfies T\left(\mathbf{u}\right) = \alpha \mathbf{u} and \mathbf{u}\left(\alpha \right) = 0 . This completes the proof.
In this section, we are interested in studying hypersurfaces of the Euclidean space R^{m+1} , which admit a unit KGVF. Let N be an orientable hypersurface of the Euclidean space R^{m+1} with shape operator T , mean curvature \alpha , and \mathbf{v} be a unit KGVFon N with respect to which the shape operator T is invariant. We prove the following:
Theorem 2. A compact and connected hypersurface N of the Euclidean space R^{m+1} , m > 1 , with mean curvature \alpha and shape operator T , admits a unit KGVF \mathbf{v} such that the shape operator T is invariant under \mathbf{v} and the function \sigma = g\left(T\mathbf{v}, \mathbf{v}\right) is nonzero and satisfies
\begin{equation*} \int_{N}m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v}\right) \geq \int_{N}\left( m(m-1)\sigma ^{2}\alpha ^{2}-\left\Vert \nabla \sigma \right\Vert ^{2}\right) \mathit{\text{, }} \end{equation*} |
if and only if m is odd, m = (2n-1) , \alpha is a constant, and N is isometric to S^{2n-1}\left(\alpha ^{2}\right) .
Proof. Suppose N is a compact and connected hypersurface of the Euclidean space R^{m+1} , m > 1 , that admits a unit KGVF \mathbf{v} such that the shape operator T is invariant under \mathbf{v} and the function \sigma = g\left(T\mathbf{v}, \mathbf{v}\right) is nonzero and satisfies the condition
\begin{equation} \int_{N}m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v}\right) \geq \int_{N}\left( m(m-1)\sigma ^{2}\alpha ^{2}-\left\Vert \nabla \sigma \right\Vert ^{2}\right) . \end{equation} | (4.1) |
Define a vector field \mathbf{u} = T\mathbf{v}-\sigma \mathbf{v} ; it follows that g\left(\mathbf{u}, \mathbf{v}\right) = 0 , that is, the vector field \mathbf{u} is orthogonal to the unit KGVF \mathbf{v} . Now, using Eq (2.12) and Lemma 1, we compute
\begin{equation*} \nabla _{E}\mathbf{u} = \left( \nabla _{E}T\right) \left( \mathbf{v}\right) +T\left( \phi E\right) -E\left( \sigma \right) \mathbf{v}-\sigma \phi E\text{ , } \end{equation*} |
that is,
\begin{equation} \nabla _{E}\mathbf{u} = \phi \left( TE\right) -E\left( \sigma \right) \mathbf{v }-\sigma \phi E\text{, }\;\;\;E\in \Psi \left( N\right) . \end{equation} | (4.2) |
Taking the inner product in the above equation with the vector field \mathbf{v} and using g\left(\mathbf{u}, \mathbf{v}\right) = 0 and Eqs (2.12) and (2.14), we get
\begin{equation*} -g\left( \mathbf{u}, \phi E\right) = -E\left( \sigma \right) \text{, } \;\;\;E\in \Psi \left( N\right) \text{, } \end{equation*} |
that is,
\begin{equation} \nabla \sigma = -\phi \mathbf{u}. \end{equation} | (4.3) |
Differentiating the above equation with respect to E\in \Psi \left(N\right) and using Eqs (4.2), (2.13), and (2.14), we get
\begin{align*} \nabla _{E}\nabla \sigma & = -\left( \nabla _{E}\phi \right) \left( \mathbf{u} \right) -\phi \left( \phi \left( TE\right) -E\left( \sigma \right) \mathbf{v} -\sigma \phi E\right) \notag \\ & = -R\left( E, \mathbf{v}\right) u-\phi ^{2}\left( TE\right) +\sigma \phi ^{2}E\text{, }\;\;\;E\in \Psi \left( N\right) . \end{align*} | (4.4) |
Note that by Eqs (2.13) and (2.14), we have
\begin{equation} R\left( E, \mathbf{v}\right) \mathbf{v} = -\phi ^{2}E\text{, }\;\;\;E\in \Psi \left( N\right), \end{equation} | (4.5) |
and using it in Eq (4.4), we conclude
\begin{equation*} \nabla _{E}\nabla \sigma = -R\left( E, \mathbf{v}\right) u+R\left( TE, \mathbf{v }\right) \mathbf{v}-\sigma R\left( E, \mathbf{v}\right) \mathbf{v}\text{ , }\;\;\;E\in \Psi \left( N\right) . \end{equation*} |
Now, using \mathbf{u} = T\mathbf{v}-\sigma \mathbf{v} to plug the first and last terms in the right-hand side of the above equation, we confirm
\begin{equation*} \nabla _{E}\nabla \sigma = -R\left( E, \mathbf{v}\right) T\mathbf{v}+R\left( TE, \mathbf{v}\right) \mathbf{v}\text{, } \end{equation*} |
which, using Eq (2.2), yields
\begin{equation*} \nabla _{E}\nabla \sigma = -\left\Vert T\mathbf{v}\right\Vert ^{2}TE+\sigma T^{2}E\text{, }\;\;\;E\in \Psi \left( N\right) . \end{equation*} |
Taking the trace in the above equation and using \Delta \sigma = {div} \left(\nabla \sigma \right) , we conclude
\begin{equation*} \Delta \sigma = -m\alpha \left\Vert T\mathbf{v}\right\Vert ^{2}+\sigma \left\Vert T\right\Vert ^{2}\text{, } \end{equation*} |
that is,
\begin{equation} \sigma \Delta \sigma = -m\alpha \sigma \left\Vert T\mathbf{v}\right\Vert ^{2}+\sigma ^{2}\left\Vert T\right\Vert ^{2}. \end{equation} | (4.6) |
Using Eq (2.3), we have
\begin{equation*} \left\Vert T\mathbf{v}\right\Vert ^{2} = m\alpha g\left( T\mathbf{v}, \mathbf{v} \right) -Ric\left( \mathbf{v}, \mathbf{v}\right) = m\alpha \sigma -Ric\left( \mathbf{v}, \mathbf{v}\right), \end{equation*} |
and inserting it in Eq (4.6), gives
\begin{equation*} \sigma \Delta \sigma = -m^{2}\alpha ^{2}\sigma ^{2}+m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v}\right) +\sigma ^{2}\left\Vert T\right\Vert ^{2}. \end{equation*} |
Integrating the above equation, yields
\begin{equation*} -\int_{N}\left\Vert \nabla \sigma \right\Vert ^{2} = \int_{N}\left( -m^{2}\alpha ^{2}\sigma ^{2}+m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v}\right) +\sigma ^{2}\left\Vert T\right\Vert ^{2}\right) \text{, } \end{equation*} |
which is rearranged as
\begin{equation*} \int_{N}\sigma ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) = \int_{N}\left( m(m-1)\alpha ^{2}\sigma ^{2}-\left\Vert \nabla \sigma \right\Vert ^{2}\right) -\int_{N}m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v}\right) . \end{equation*} |
Using the inequality (4.1) in the above equation, it confirms
\begin{equation*} \int_{N}\sigma ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) \leq 0. \end{equation*} |
However, by Schwartz's inequality, we have \left\Vert T\right\Vert ^{2}\geq m\alpha ^{2} and therefore, the integrand on the left-hand side of the above inequality is non-negative. Hence, we have
\begin{equation*} \sigma ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) = 0, \end{equation*} |
with the function \sigma nonzero on connected N , which implies \left(\left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) = 0 . The equality \left\Vert T\right\Vert ^{2} = m\alpha ^{2} in Schwartz's inequality holds if and only if
\begin{equation} T = \alpha I\text{, } \end{equation} | (4.7) |
which gives
\begin{equation*} \left( \nabla _{E}T\right) (F) = E\left( \alpha \right) F\text{, }\;\;\;E, F\in \Psi \left( N\right) . \end{equation*} |
Taking a local frame \left\{ w_{k}\right\} _{1}^{m} on the hypersurface N , in the above equation, we have
\begin{equation*} \sum\limits_{k = 1}^{m}\left( \nabla _{w_{k}}T\right) (w_{k}) = \sum\limits_{k = 1}^{m}w_{k}\left( \alpha \right) w_{k}\text{, } \end{equation*} |
which, in view of Eq (2.7), implies
\begin{equation*} m\nabla \alpha = \nabla \alpha, \end{equation*} |
and as m > 1 , it confirms that \alpha is a constant. Then, by Eqs (2.2) and (4.7), we have
\begin{equation*} R\left( E, F\right) G = \alpha ^{2}\left\{ g\left( F, G\right) E-g\left( E, G\right) F\right\} \text{, }\;\;\;E, F, G\in \Psi \left( N\right) . \end{equation*} |
Note that \alpha \neq 0 , because compact minimal hypersurfaces in Euclidean space do not exist. Hence, \alpha ^{2} > 0 , and N is isometric to S^{m}\left(\alpha ^{2}\right) . Note that a Killing vector field on an even-dimensional compact Riemannian manifold of positive sectional curvature must vanish at some point [11]. Therefore, as \mathbf{v} is a unit vector field, it never vanishes, and it announces that m cannot be even. Hence, m = 2n-1 , that is, N is isometric to S^{2n-1}\left(\alpha ^{2}\right) .
Conversely, suppose N is isometric to S^{2n-1}\left(\alpha ^{2}\right) . Then by Eqs (1.3) and (1.4), there is a unit vector field \mathbf{v} = J\zeta on S^{2n-1}\left(\alpha ^{2}\right) that satisfies
\begin{equation} \nabla _{E}\mathbf{v} = \alpha \left( JE\right) ^{T}\text{, }\;\;\;E\in \Psi \left( S^{2n-1}\left( \alpha ^{2}\right) \right) \text{, } \end{equation} | (4.8) |
where J is the complex structure of the ambient Euclidean space R^{2n} , and \zeta is the unit normal, and \left(JE\right) ^{T} is the tangential projection of the vector field JE to S^{2n-1}\left(\alpha ^{2}\right) . Taking the inner product in Eq (4.8) by the vector field F on the sphere S^{2n-1}\left(\alpha ^{2}\right) , we have
\begin{equation*} g\left( \nabla _{E}\mathbf{v}, F\right) = \alpha g\left( \left( JE\right) ^{T}\left( JE\right) ^{T}, F\right) = \alpha \langle JE, F\rangle \text{, } \end{equation*} |
and we conclude
\begin{equation*} \left( \mathtt{£} _{\mathbf{v}}g\right) \left( E, F\right) = \alpha \langle JE, F\rangle +\alpha \langle JF, E\rangle = 0, \end{equation*} |
by virtue of the skew symmetry of the complex structure, that is, the Euclidean metric is a Hermitian metric. Hence, \mathbf{v} is a unit KGVF on S^{2n-1}\left(\alpha ^{2}\right) . Note that, in this case the shape operator is T = \alpha I , and the function \sigma = g\left(T\mathbf{v}, \mathbf{v}\right) = \alpha is a nonzero constant. Moreover, with m = 2n-1
\begin{equation} \int_{S^{2n-1}\left( \alpha ^{2}\right) }m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v}\right) = \int_{S^{2n-1}\left( \alpha ^{2}\right) }2(2n-1)(n-1)\alpha ^{4} \end{equation} | (4.9) |
and
\begin{equation} \int_{S^{2n-1}\left( \alpha ^{2}\right) }\left( m(m-1)\sigma ^{2}\alpha ^{2}-\left\Vert \nabla \sigma \right\Vert ^{2}\right) = \int_{S^{2n-1}\left( \alpha ^{2}\right) }2(2n-1)(n-1)\alpha ^{4}, \end{equation} | (4.10) |
as \nabla \sigma = 0 . Hence, by Eqs (4.9) and (4.10), we get
\begin{equation*} \int_{S^{2n-1}\left( \alpha ^{2}\right) }m\alpha \sigma Ric\left( \mathbf{v}, \mathbf{v}\right) = \int_{S^{2n-1}\left( \alpha ^{2}\right) }\left( m(m-1)\sigma ^{2}\alpha ^{2}-\left\Vert \nabla \sigma \right\Vert ^{2}\right), \end{equation*} |
and this finishes the proof.
Let N be an immersed hypersurface in the Euclidean space R^{m+1} with unit normal \zeta , shape operator T , and mean curvature \alpha . Let \varphi :N\rightarrow R^{m+1} be the immersion and \rho = \langle \varphi, \zeta \rangle be the support of N . The position vector field \varphi is expressed as
\begin{equation} \varphi = \mathbf{u}+\rho \zeta, \end{equation} | (5.1) |
and we call \mathbf{u} the basic vector field of the hypersurface N . Differentiating Eq (5.1), using Eq (2.1), and equating similar components, we get
\begin{equation} \nabla _{E}\mathbf{u} = E+\rho TE\text{, }\;\;\;\nabla \rho = -T\mathbf{u}\text{ , }\;\;\;E\in \Psi (N). \end{equation} | (5.2) |
The first equation in Eq (5.2), gives
\begin{equation} {div}\mathbf{u} = m\left( 1+\rho \alpha \right) . \end{equation} | (5.3) |
In this section, we prove the following result:
Theorem 3. A compact and connected immersed hypersurface N of the Euclidean space R^{m+1} , m > 1 , with nonzero support \rho and basic vector field \mathbf{u} satisfies _{{}}
\begin{equation*} \int_{N}Ric\left( \mathbf{u}, \mathbf{u}\right) \geq \frac{m-1}{m} \int_{N}\left( {div}\mathbf{u}\right) ^{2}\mathit{\text{, }} \end{equation*} |
if and only if, the mean curvature \alpha is a constant and N is isometric to S^{m}(\alpha ^{2}) .
Proof. Suppose that the immersed hypersurface N of the Euclidean space R^{m+1} , m > 1 , has nonzero support \rho and the basic vector field \mathbf{u} satisfy
\begin{equation} \int_{N}Ric\left( \mathbf{u}, \mathbf{u}\right) \geq \frac{m-1}{m} \int_{N}\left( {div}\mathbf{u}\right) ^{2}. \end{equation} | (5.4) |
Using Eq (5.2), we have
\begin{equation*} \rho \left( TE-\alpha E\right) = \nabla _{E}\mathbf{u}-\left( 1+\rho \alpha \right) E, \end{equation*} |
and using a local frame \left\{ w_{k}\right\} _{1}^{m} on the hypersurface N with the above equation, we get
\begin{eqnarray*} \rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2} & = &\sum\limits_{k = 1}^{m}g\left( \rho \left( Tw_{k}-\alpha w_{k}\right) , \rho \left( Tw_{k}-\alpha w_{k}\right) \right) \\ & = &\sum\limits_{k = 1}^{m}g\left( \nabla _{w_{k}}\mathbf{u}-\left( 1+\rho \alpha \right) w_{k}, \nabla _{w_{k}}\mathbf{u}-\left( 1+\rho \alpha \right) w_{k}\right) \\ & = &\left\Vert \nabla \mathbf{u}\right\Vert ^{2}+m\left( 1+\rho \alpha \right) ^{2}-2\left( 1+\rho \alpha \right) {div}\mathbf{u}. \end{eqnarray*} |
Using Eq (5.3) in the above equation, we have
\begin{equation} \rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = \left\Vert \nabla \mathbf{u} \right\Vert ^{2}-\frac{1}{m}\left( {div}\mathbf{u}\right) ^{2}. \end{equation} | (5.5) |
Note that on using Eq (5.2), we have
\begin{equation*} \left( \mathtt{£} _{\mathbf{u}}g\right) \left( E, F\right) = 2g\left( E, F\right) +2\rho g\left( TE, F\right) \text{, }\;\;\;E, F\in \Psi (N)\text{, } \end{equation*} |
which gives
\begin{eqnarray*} \left\vert \mathtt{£} _{\mathbf{u}}g\right\vert ^{2} & = &\sum\limits_{jk}\left( \mathtt{£} _{\mathbf{u}}g\right) \left( w_{j}, w_{k}\right) = 4\sum\limits_{jk}\left( g\left( w_{j}, w_{k}\right) +\rho g\left( Tw_{j}, w_{k}\right) \right) ^{2} \\ & = &4\left( m+2m\rho \alpha +\rho ^{2}\left\Vert T\right\Vert ^{2}\right) . \end{eqnarray*} |
Integrating the last equation, while using Minkowski's formula, we have
\begin{equation} \frac{1}{2}\int_{N}\left\vert \mathtt{£} _{\mathbf{u}}g\right\vert ^{2} = 2\int_{N}\left( \rho ^{2}\left\Vert T\right\Vert ^{2}+m\rho \alpha \right) . \end{equation} | (5.6) |
Next, we recall the following integral formula [20]
\begin{equation*} \int_{N}\left( Ric\left( \mathbf{u}, \mathbf{u}\right) +\frac{1}{2} \left\vert \mathtt{£} _{\mathbf{u}}g\right\vert ^{2}-\left\Vert \nabla \mathbf{u }\right\Vert ^{2}-\left( {div}\mathbf{u}\right) ^{2}\right) = 0, \end{equation*} |
which holds for any vector field on the compact Riemannian manifold (N, g).
Using the above integral formula with the integral of Eq (5.5), we get
\begin{equation} \int_{N}\rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = \int_{N}\left( Ric\left( \mathbf{u}, \mathbf{u}\right) +\frac{1}{ 2}\left\vert \mathtt{£} _{\mathbf{u}}g\right\vert ^{2}-\left( {div}\mathbf{ u}\right) ^{2}-\frac{1}{m}\left( {div}\mathbf{u}\right) ^{2}\right) . \end{equation} | (5.7) |
Now, using Eq (1.1) in Eq (5.6), we have
\begin{eqnarray*} \frac{1}{2}\int_{N}\left\vert \mathtt{£} _{\mathbf{u}}g\right\vert ^{2} & = &2\int_{N}\left( \rho ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) +m\left( \rho ^{2}\alpha ^{2}+\rho \alpha \right) \right) \\ & = &2\int_{N}\left( \rho ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) +m\left( \rho ^{2}\alpha ^{2}+2\rho \alpha +1\right) \right) \\ & = &2\int_{N}\left( \rho ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) +m\left( 1+\rho \alpha \right) ^{2}\right) . \end{eqnarray*} |
Employing (5.1), in the above equation, we conclude
\begin{equation*} \frac{1}{2}\int_{N}\left\vert \mathtt{£} _{\mathbf{u}}g\right\vert ^{2} = 2\int_{N}\left( \rho ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) +\frac{1}{m}\left( {div}\mathbf{u}\right) ^{2}\right) . \end{equation*} |
Inserting this equation in Eq (5.7), we find that
\begin{equation} \int_{N}\rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = \int_{N}\left( Ric\left( \mathbf{u}, \mathbf{u}\right) +2\rho ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) -\frac{m-1}{m} \left( {div}\mathbf{u}\right) ^{2}\right) . \end{equation} | (5.8) |
Finally, observe that
\begin{eqnarray*} \left\Vert T-\alpha I\right\Vert ^{2} & = &\sum\limits_{k = 1}^{m}g\left( Tw_{k}-\alpha w_{k}, Tw_{k}-\alpha w_{k}\right) \\ & = &\left\Vert T\right\Vert ^{2}-2m\alpha ^{2}+m\alpha ^{2}\text{, } \end{eqnarray*} |
that is,
\begin{equation*} \rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = \rho ^{2}\left( \left\Vert T\right\Vert ^{2}-m\alpha ^{2}\right) \end{equation*} |
and utilizing the above equation in Eq (5.8), we obtain
\begin{equation*} \int_{N}\rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = \frac{m-1}{m} \int_{N}\left( {div}\mathbf{u}\right) ^{2}-\int_{N}Ric\left( \mathbf{u}, \mathbf{u}\right) . \end{equation*} |
Using inequality (5.4) in the above equation, we get
\begin{equation*} \int_{N}\rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2}\leq 0\text{, } \end{equation*} |
which gives \rho ^{2}\left\Vert T-\alpha I\right\Vert ^{2} = 0 . However, the support \rho \neq 0 on connected N implies
\begin{equation*} T = \alpha I, \end{equation*} |
and as in the proof of Theorem 2, we realize that \alpha is a constant, and by Eq (2.2), the curvature tensor of N is given by
\begin{equation*} R\left( E, F\right) G = \alpha ^{2}\left\{ g\left( F, G\right) E-g\left( E, G\right) F\right\} \text{, }\;\;\;E, F, G\in \Psi \left( N\right) \text{, } \end{equation*} |
with constant \alpha \neq 0 as there are no compact minimal hypersurfaces in the Euclidean space. Hence, N is isometric to S^{m}\left(\alpha ^{2}\right) .
Conversely, suppose N is isometric to S^{m}\left(\alpha ^{2}\right) . Then, the embedding \varphi :S^{m}\left(\alpha ^{2}\right) \rightarrow R^{m+1} has shape operator T = \alpha I , unit normal \zeta = -\alpha \varphi and support \rho = -\frac{1}{\alpha }\neq 0 . Moreover, the basic vector field \mathbf{u} = 0 . Hence, the condition (5.4) vacuously holds as an equality.
In Sections 3 and 4, we have employed a CLVF and a KGVF on a compact hypersurface N , respectively, of the Euclidean space R^{m+1} to find a characterization of spheres S^{m}\left(c\right) and S^{2n-1}(c) , respectively. This further increases the scope of the study of hypersurfaces in the Euclidean space R^{m+1} ; for instance, one would be interested in analyzing the impact of the presence of a geodesic vector field \xi on an orientable hypersurface N of the Euclidean space R^{m+1} [10]. A vector field \xi on a Riemannian manifold \left(N, g\right) is said to be a geodesic vector field, if its integral curves are geodesics of \left(N, g\right) . A unit Killing vector field on \left(N, g\right) is a geodesic vector field, and the converse is not true. To support this fact that a geodesic vector field need not be a KGVF, we need to introduce a 3 -dimensional trans-Sasakian manifold \left(N, g, \phi, \zeta, \eta, f, h\right) , where (N, g) is a 3 -dimensional Riemannian manifold, \phi is a (1, 1) tensor field, \zeta is a unit vector field (called Reeb vector field), \eta is 1 -form dual to \zeta , and f , h are smooth functions on M satisfying [1]
\phi ^{2} = -I+\eta \otimes \zeta \text{, }\;\;\;\phi \left( \zeta \right) = 0 \text{, }\;\;\;\eta \circ \phi = 0\text{, }\;\;\;g\left( \phi E, \phi F\right) = g\left( E, F\right) -\eta \left( E\right) \eta \left( F\right) |
and
\begin{eqnarray*} \nabla _{E}\zeta & = &-f\phi E+h\left( E-\eta (E)\zeta \right) \text{, } \\ \left( \nabla _{E}\phi \right) (F) & = &f\left( g\left( E, F\right) \xi -\eta \left( F\right) E\right) +h\left( g\left( \phi E, F\right) \xi -\eta \left( F\right) \phi E\right) , \end{eqnarray*} |
E, F\in \Psi (N) . A trans-Sasakian manifold \left(N, g, \phi, \zeta, \eta, f, h\right) is said to be proper, if neither of the functions f nor h are zero. It is easy to see that \nabla _{\zeta }\zeta = 0 , that is, \zeta is a geodesic vector field. However, on a proper trans-Sasakian manifold \left(N, g, \phi, \zeta, \eta, f, h\right)
\left( \mathtt{£} _{\zeta }g\right) \left( E, F\right) = 2hg\left( \phi E, \phi F\right) \neq 0\text{, } |
that is, \zeta is not a Killing vector field. Hence, on a proper trans-Sasakian manifold \left(N, g, \phi, \zeta, \eta, f, h\right) , the Reeb vector field \zeta is a geodesic vector field that is not a KGVF. Thus, a geodesic vector field being a nontrivial generalization of a Killing vector field makes it a potential case for studying the impact of the presence of a geodesic vector field on the geometry of an orientable hypersurface of the Euclidean space R^{m+1} .
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to extend their sincere appreciation to Supporting Project number (RSPD2024R860) King Saud University, Riyadh, Saudi Arabia.
The authors declare that they have no conflicts of interest.
[1] |
I. Al-Dayel, S. Deshmukh, G. Vîlcu, Trans-Sasakian static spaces, Results Phys., 31 (2021), 105009. https://doi.org/10.1016/j.rinp.2021.105009 doi: 10.1016/j.rinp.2021.105009
![]() |
[2] |
V. Berestovskii, Y. Nikonorov, Killing vector fields of constant length on Riemannian manifolds, Siberian Math. J., 49 (2008), 395–407. https://doi.org/10.48550/arXiv.math/0605371 doi: 10.48550/arXiv.math/0605371
![]() |
[3] |
B. Y. Chen, A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relat. Gravit., 46 (2014), 1833. https://doi.org/10.48550/arXiv.1411.0270 doi: 10.48550/arXiv.1411.0270
![]() |
[4] |
B. Y. Chen, Some results on concircular vector fields and their applications to Ricci solitons, Bull. Korean Math. Soc., 52 (2015), 1535–1547. https://doi.org/10.4134/BKMS.2015.52.5.1535 doi: 10.4134/BKMS.2015.52.5.1535
![]() |
[5] |
B. Y. Chen, S. Deshmukh, Some results about concircular vector fields on Riemannian manifolds, Filomat, 34 (2020), 835–842. https://doi.org/10.2298/FIL2003835C doi: 10.2298/FIL2003835C
![]() |
[6] |
D. Chen, C. X. Wang, X. S. Wang, A characterization of separable hypersurfaces in euclidean space, Math. Notes, 113 (2023), 339–344. https://doi.org/10.1134/S0001434623030033 doi: 10.1134/S0001434623030033
![]() |
[7] |
S. Deshmukh, Compact hypersurfaces in a Euclidean space, Q. J. Math., 49 (1998), 35–41. https://doi.org/10.1093/qmathj/49.1.35 doi: 10.1093/qmathj/49.1.35
![]() |
[8] |
S. Deshmukh, A note on spheres in a Euclidean space, Publ. Math. Debrecen, 64 (2004), 31–37. https://doi.org/10.5486/PMD.2004.2843 doi: 10.5486/PMD.2004.2843
![]() |
[9] |
S. Deshmukh, An integral formula for compact hypersurfaces in a Euclidean space and its applications, Glasgow Math. J., 34 (1992), 309–311. https://doi.org/10.1017/S0017089500008867 doi: 10.1017/S0017089500008867
![]() |
[10] |
S. Deshmukh, V. A. Khan, Geodesic vector fields and eikonal equation on a Riemannian manifold, Indag. Math., 30 (2019), 542–552. https://doi.org/10.1016/j.indag.2019.02.001 doi: 10.1016/j.indag.2019.02.001
![]() |
[11] | M. D. Carmo, Riemannian Geometry, Birkhäuser, 1992. https://doi.org/10.2307/3618122 |
[12] |
T. Hasanis, R. López, Classification of separable surfaces with constant Gaussian curvature, Manuscript Math., 166 (2021), 403–417. https://doi.org/10.48550/arXiv.1912.07870 doi: 10.48550/arXiv.1912.07870
![]() |
[13] |
T. Hasanis, R. López, Translation surfaces in Euclidean space with constant Gaussian curvature, Commun. Anal. Geom., 29 (2021), 1415–1447. https://doi.org/10.48550/arXiv.1809.02758 doi: 10.48550/arXiv.1809.02758
![]() |
[14] |
W. C. Lynge, Sufficient conditions for periodicity of a Killing vector field, Proc. Amer. Math. Soc., 38 (1973), 614–616. https://doi.org/10.2307/2038961 doi: 10.2307/2038961
![]() |
[15] | M. Obata, Conformal transformations of Riemannian manifolds, J. Diff. Geom., 4 (1970), 311–333. |
[16] |
M. Obata, The conjectures about conformal transformations. J. Diff. Geom., 6 (1971), 247–258. https://doi.org/10.4310/jdg/1214430407 doi: 10.4310/jdg/1214430407
![]() |
[17] |
X. Rong, Positive curvature, local and global symmetry, and fundamental groups, Amer. J. Math., 121 (1999), 931–943. https://doi.org/10.1353/ajm.1999.0036 doi: 10.1353/ajm.1999.0036
![]() |
[18] |
G. Ruiz-Hernández, Translation hypersurfaces whose curvature depends partially on its variables, J. Math. Anal. Appl., 479 (2021), 124913. https://doi.org/10.1016/j.jmaa.2020.124913 doi: 10.1016/j.jmaa.2020.124913
![]() |
[19] |
D. D. Saglam, C. Sunar, Translation hypersurfaces of semi-Euclidean spaces with constant scalar curvature, AIMS Math., 8 (2022), 5036–5048. https://doi.org/10.3934/math.2023252 doi: 10.3934/math.2023252
![]() |
[20] | K. Yano, Integral formulas in riemannian geometry, New York, 1970. https://doi.org/10.1017/S0008439500031520 |
[21] |
S. Yorozu, Killing vector fields on complete Riemannian manifolds, Proc. Amer. Math. Soc., 84 (1982), 115–120. https://doi.org/10.2307/2043822 doi: 10.2307/2043822
![]() |