A Note on Shape Vector Fields on Hypersurfaces

Abstract

In this paper, we initiate the study of shape vector fields on the hypersurfaces of a Riemannian manifold. We use a shape vector field on a compact hypersurface of a Euclidean space to obtain a characterization of round spheres. We also find a condition, under which a shape vector field that is on a compact hypersurface of a Euclidean space is a Killing vector field.

1. Introduction

In the geometry of the hypersurfaces of a Euclidean space, the main tools (as the functions), are defined on a hypersurface, and these are the mean curvature $\alpha$, scalar curvature $\tau$, the squared norms of the shape operator S, the Ricci operator Q and the curvature tensor field R. In [1], it was shown that a compact hypersurface M of the Euclidean space $R^{n+1}$ satisfying the inequality

$${∥S∥}^2\tau \ge \frac{1}{2}{∥R∥}^2+{∥Q∥}^2+2n(n-1){∥\nabla \alpha ∥}^2$$

is necessarily isometric to the n-sphere $S^n\left({\alpha }^2\right)$. In [2], the position vector field T of a compact hypersurface M in the Euclidean space $R^{n+1}$ was used to define a smooth vector field $\xi$ on the hypersurface M, namely that $\xi$ is the tangential component of the position vector field T, which is required to obtain the following integral formula

$${\int }_M\left\{Ric(\xi ,\xi )+n(n-1)-{\rho }^2\tau \right\}=0,$$

where $\rho$ is the support function of the hypersurface. This integral formula leads to several important results on the geometry of the compact hypersurface M of the Euclidean space $R^{n+1}$ (cf. [2]). There is also another aspect in the study of hypersurfaces of a Euclidean space, namely the existence of special vector fields on a hypersurface, such as a conformal vector field or a Killing vector field. In [3], it was shown that a compact hypersurface M of the Euclidean space $R^{n+1}$ admits a Killing vector field unit $\xi$, such that the function $g\left(S\xi ,\xi \right)$ is a constant and is, necessarily, an odd dimensional sphere of a constant curvature.

There is yet another type of hypersurface in the Euclidean space $R^{n+1}$, namely the translation hypersurface. In this type of hypersurface, the focus is on the analysis of the smooth function $f:R^{n+1}\to R$, whose graph is the desired hypersurface under study. For instance, in [4], the author investigates translation hypersurfaces in the Euclidean space $R^{n+1}$, i.e., those which have a Gauss–Kronecker curvature depending on either its first p variables or on the rest of the q variables. These hypersurfaces are graphs of a function, i.e., the sum of two functions in the p and q variables, whereby $n=p+q$. In [4], the authors found conditions on a translation hypersurface that had a Gauss–Kronecker curvature of zero. As generalizations of translation hypersurfaces, the notion of separable hypersurfaces in the Euclidean space $R^{n+1}$ has also been introduced (cf. [5,6,7,8]). Regarding separable hypersurfaces, the defining smooth function, whose graph is the translation hypersurface, is required to satisfy the additional conditions. In [8], the authors studied separable hypersurfaces in $R^{n+1}$, as well as a vanishing Gauss–Kronecker curvature.

We should recall that specific types of vector fields, such as Killing and conformal vector fields, as well as differential equations, like Obata’s and Fischer–Marsden’s, have a great significance in characterizing important Riemannian manifolds (cf. [1,2,3,9,10,11,12,13,14,15]). On a Riemannian manifold $(M,g)$, a conformal vector field $\xi$—which is defined as a vector field for which the local flow corresponds to conformal transformations of $(M,g)$—is equivalently the following:

$$\frac{1}{2}{\mathcal{L}}_{\xi }g=\sigma g,$$

where the Lie derivative of the metric g with respect to the vector field $\xi$ is represented as ${\mathcal{L}}_{\xi }g$, while $\sigma$ denotes a smooth function on M (which is referred to as the conformal factor associated with $\xi$). If $\sigma =0$, then $\xi$ is said to be a Killing vector field. For M, an orientable hypersurface of the Riemannian manifold $(\overline{M},\overline{g})$ is equipped with a shape operator S and a unit normal N. If we define a smooth vector field $\xi$ on the hypersurface M, it is called a shape vector field if it meets the following conditions:

$$\frac{1}{2}{\mathcal{L}}_{\xi }g=\sigma B,$$

(1)

where B is defined on M by

$$B(U,V)=g\left(SU,V\right),U,V\in \mathfrak{X}\left(M\right),$$

and the space of vector fields on the hypersurface M is denoted as $\mathfrak{X}\left(M\right)$. The potential function of the shape vector field $\xi$ defined on the hypersurface M is denoted by the smooth function $\sigma$. In the case of the unit sphere ${\mathbf{S}}^{2n+1}$, as a hypersurface of Euclidean space ${\mathbf{R}}^{2(n+1)}$, the Reeb vector field $\xi$ on ${\mathbf{S}}^{2n+1}$ is a shape vector field with a potential function $\sigma =0$. We will see that there are shape vector fields on the sphere ${\mathbf{S}}^n\left(c\right)$, which have a constant curvature c as the hypersurfaces of the Euclidean space ${\mathbf{R}}^{n+1}$ (which also have a nonzero potential functions, as shown in Section 3).

Consider an orientable hypersurface M in ${\mathbf{R}}^{n+1}$ that is characterized by a unit normal vector N and a shape operator S. Let $\xi$ be a shape vector field on M with a corresponding potential function $\sigma$. We will use $\eta$ to denote the 1-form that is dual to the shape vector field $\xi$, that is,

$$\eta \left(U\right)=g\left(U,\xi \right),U\in \mathfrak{X}\left(M\right),$$

where g is the induced metric on M. Then, the operator F is defined on the hypersurface M by

$$\frac{1}{2}d\eta \left(U,V\right)=g\left(FU,V\right),U,V\in \mathfrak{X}\left(M\right),$$

(2)

such that $d\eta$ is the exterior derivative of $\eta$. We call F the operator associated with the shape vector field. For a local orthonormal frame $\left\{v_1,\dots ,v_n\right\}$, which is defined on the hypersurface, the squared length of F is represented as

$${∥F∥}^2=\sum _{i=1}^{n}g\left(F{v}_i,F{v}_i\right).$$

We denote the above by $\alpha$ as the mean curvature of the hypersurface M, and this is defined by

$$n\alpha =Tr·S,$$

where $Tr·S$ is the trace of S. In this paper, we employed the shape vector field that was defined on the hypersurface, with a constant curvature c, to establish the subsequent characterization of the sphere ${\mathbf{S}}^n\left(c\right)$.

Theorem 1.

Let M be compact and simply connected to the hypersurface of ${\mathbf{R}}^{n+1}$, $n\ge 2$, with mean curvature α, and let ξ be a shape vector field with a nonzero potential function σ, as well as with the associated operator F on M. Then, the Ricci curvature $Ric(\xi ,\xi )$ satisfies

$${\int }_{M}Ric\left(\xi ,\xi \right)\ge {\int }_M\left(n(n-1){\sigma }^2{\alpha }^2+{∥F∥}^2\right),$$

if and only if α is a constant and M is isometric to the sphere ${\mathbf{S}}^n\left({\alpha }^2\right)$.

In the methods of the proof of Theorem 1, we found that, due to the restriction on the Ricci curvature, the shape vector field $\xi$ turned out to be a conformal vector field. Thus, a natural question arises: can a shape vector field on the hypersurface of a Euclidean space be a Killing vector field? In the next result, we answer this question and determine the conditions that arise when a shape vector field on a compact hypersurface of a Euclidean space is a Killing vector field.

Theorem 2.

Let M be a compact and connected hypersurface with a nonzero scalar curvature $\mathbf{ø}$ of ${\mathbf{R}}^{n+1}$, and let ξ be a shape vector field with potential function σ on M. If the function $\sigma \mathbf{ø}$ does not change sign on M, then ξ will be a Killing vector field.

Now, let us discuss an example of a shape vector field on a hypersurface within the Euclidean space $R^{n+1}$. The geometry of Einstein hypersurfaces in a Euclidean space is very rich (cf. [16,17]). Thus, let M be a non-total geodesic that is a completely umbilical Einstein hypersurface of the Euclidean space $R^{n+1}$. Then, we will obtain the scalar curvature $\tau$ of the hypersurface, which is a constant and has a Ricci curvature that is given by

$$Ric=\frac{\tau }{n}g.$$

(3)

The Fischer–Marsden equation is considered one of the most significant differential equations on a Riemannian manifold $(M,g)$, and it is given by

$$\left(\Delta \rho \right)g+\rho Ric=Hess\left(g\right)$$

(4)

We shall call the above equation the FM-equation for convenience (cf. [18]). It has been widely acknowledged that, in a Riemannian manifold $(M,g)$, the scalar curvature $\tau$ admits a nontrivial solution of FM-equation that is a constant (cf. [18]). Indeed, it has been conjectured that a compact $(M,g)$ has a nontrivial solution $\rho$ for the FM-Equation (4), which should imply that $(M,g)$ is an Einstein manifold. However, this conjecture has turned out to be false. This is due to the fact that it is well known that the FM-equation on a Riemannian manifold facilitates the study of its geometry (cf. [19,20,21]). As such, let M be a non-total geodesic that is a completely umbilical Einstein hypersurface of the Euclidean space $R^{n+1}$, wherein a nontrivial solution $\rho$ of the FM-equation is admitted. Then, by undertaking a trace in Equation (4), we can obtain

$$\Delta \rho =-\frac{\tau }{n-1}\rho .$$

(5)

By taking $\xi =\nabla \rho$ on the hypersurface M, i.e., when using Equation (4), we can conclude

$$g\left({\nabla }_U\xi ,V\right)=-\frac{\tau }{n-1}\rho g\left(U,V\right)+\frac{\tau }{n}\rho g\left(U,V\right)=-\frac{\tau }{n(n-1)}\rho g\left(U,V\right).$$

(6)

Since M is non-total geodesic completely umbilical hypersurface, we can obtain $SU=\alpha U$, the constant $\alpha \ne 0$ and—consequently—Equation (6), which implies that

$$\left({\pounds }_{\xi }g\right)\left(U,V\right)=-\frac{2\tau }{n(n-1)\alpha }\rho g\left(AU,V\right),$$

that is,

$$\frac{1}{2}{\pounds }_{\xi }g=\sigma B,\sigma =-\frac{\tau }{n(n-1)\alpha }\rho .$$

Hence, on the hypersurface M of the Euclidean space $R^{n+1}$, $\xi$ is a shape vector field.

2. Preliminaries

For an orientable hypersurface M of the Euclidean space ${\mathbf{R}}^{n+1}$ (with a unit normal vector N and a shape operator S), we can denote this by $⟨,⟩$ for the Euclidean metric. In addition, we can denote the induced metric on M by g, as well as denote $\overline{\nabla }$ and ∇ by the Euclidean connection and the Riemannian connection on ${\mathbf{R}}^{n+1}$ and M, respectively. Then, for the hypersurface M, the fundamental equations are as follows:

$${\overline{\nabla }}_{U}V={\nabla }_{U}V+g\left(SU,V\right)N,U,V\in \mathfrak{X}\left(M\right)$$

(7)

and

$${\overline{\nabla }}_{U}N=-SU,U\in \mathfrak{X}\left(M\right).$$

(8)

For the hypersurface M, the curvature tensor R is given by

$$R\left(U,V\right)W=g\left(SV,W\right)SU-g\left(SU,W\right)SV,U,V,W\in \mathfrak{X}\left(M\right),$$

(9)

For the hypersurface M, the Ricci tensor is given by

$$Ric\left(U,V\right)=n\alpha g\left(SU,V\right)-g\left(SU,SV\right),U,V\in \mathfrak{X}\left(M\right),$$

(10)

where the hypersurface M and the mean curvature $\alpha$ is denoted by $n\alpha =Tr·S$. In addition, the Ricci operator Q is defined by

$$Ric\left(U,V\right)=g\left(QU,V\right),U,V\in \mathfrak{X}\left(M\right)$$

(11)

and the scalar curvature $\mathbf{ø}$ is introduced by

$$\mathbf{ø}=Tr·Q.$$

(12)

Using Equation (10), we can obtain

$$QU=n\alpha SU-S^{2}U,\mathbf{ø}=n^2{\alpha }^2-{∥S∥}^2,$$

(13)

where

$${∥S∥}^2=\sum _{i=1}^{n}g\left(S{v}_i,S{v}_i\right)$$

and $\left\{v_1,\dots ,v_n\right\}$ is a local frame on the hypersurface M. As the Euclidean space ${\mathbf{R}}^{n+1}$ is flat, the Codazzi equation for the hypersurface has the form

$$\left(\nabla S\right)\left(U,V\right)=\left(\nabla S\right)\left(V,U\right),U,V\in \mathfrak{X}\left(M\right),$$

(14)

where

$$\left(\nabla S\right)\left(U,V\right)={\nabla }_{U}SV-S{\nabla }_{U}V.$$

Also, by considering a local frame $\left\{v_1,\dots ,v_n\right\}$ on the hypersurface M, and by using $n\alpha =Tr·S$, i.e.,

$$n\alpha =\sum _{i=1}^{n}g\left(S{v}_i,v_i\right),$$

and then using the symmetry of the shape operator S, as well as Equation (14), we can compute

$$nU\left(\alpha \right)=g\left(U,\sum _{i=1}^n\left(\nabla S\right)\left(v_i,v_i\right)\right),U\in \mathfrak{X}\left(M\right).$$

Thus, the mean curvature $\alpha$ has the gradient

$$\nabla \alpha =\frac{1}{n}\sum _{i=1}^n\left(\nabla S\right)\left(v_i,v_i\right).$$

(15)

Regarding the hypersurface M, consider $\xi$ as the shape vector field with the potential function $\sigma$ and the associated operator F. Then, we can obtain

$$\frac{1}{2}{\mathcal{L}}_{\xi }g=\sigma B,\frac{1}{2}d\eta \left(U,V\right)=g\left(FU,V\right),U,V\in \mathfrak{X}\left(M\right),$$

where $B(U,V)=g\left(SU,V\right)$ and $\eta$ are the 1-form that is dual to $\xi$. Then, by applying Koszul’s formula (cf. [10]), we can obtain

$${\nabla }_U\xi =\sigma SU+FV,U\in \mathfrak{X}\left(M\right).$$

(16)

When using the formula of the curvature tensor

$$R\left(U,V\right)W={\nabla }_U{\nabla }_{V}W-{\nabla }_V{\nabla }_{U}W-{\nabla }_{[U,V]}W,U,V,W\in \mathfrak{X}\left(M\right),$$

and Equations (14) and (16), we can confirm

$$R\left(U,V\right)\xi =U\left(\sigma \right)SV-V\left(\sigma \right)SU+\left(\nabla F\right)(U,V)-\left(\nabla F\right)\left(V,U\right),U,V\in \mathfrak{X}\left(M\right).$$

(17)

Regarding the hypersurface M, when using a local frame $\left\{v_1,\dots ,v_n\right\}$ and the following formula of the Ricci tensor

$$Ric\left(U,\xi \right)=\sum _{i=1}^{n}g\left(R\left(v_i,U\right)\xi ,v_i\right)$$

as well as Equation (17), we arrive at

$$Ric\left(U,\xi \right)=g\left(S\nabla \sigma ,U\right)-n\alpha U\left(\sigma \right)+\sum _{i=1}^{n}g\left(\left(\nabla F\right)(v_i,U),v_i\right)-\sum _{i=1}^{n}g\left(\left(\nabla F\right)(U,v_i),v_i\right),$$

In this context, the symmetry of the shape operator S and the equation $n\alpha =\mathrm{Tr}\left(S\right)$ have been utilized. Now, by applying the skew symmetry property of the operator F, its outcomes $Tr.F=0$ and

$$g\left(\left(\nabla F\right)(U,V),W\right)=-g\left(V,\left(\nabla F\right)\left(U,W\right)\right),$$

we can conclude

$$Ric\left(U,\xi \right)=g\left(S\nabla \sigma ,U\right)-n\alpha U\left(\sigma \right)-\sum _{i=1}^{n}g\left(U,\left(\nabla F\right)(v_i,v_i)\right),U\in \mathfrak{X}\left(M\right),$$

that is,

$$Ric\left(\xi ,\xi \right)=\left(S\xi \right)\left(\sigma \right)-n\alpha \xi \left(\sigma \right)-\sum _{i=1}^{n}g\left(\xi ,\left(\nabla F\right)(v_i,v_i)\right).$$

(18)

3. Proof of Theorem 1

Regarding M, there is a hypersurface of ${\mathbf{R}}^{n+1}$, $n\ge 2$, which is compact and simply connected with a unit normal vector N and a shape operator S, where a shape vector field $\xi$ that has a nonzero potential function $\sigma$ and a related operator F is admitted. Suppose the Ricci curvature $Ric\left(\xi ,\xi \right)$ of hypersurface M satisfies

$${\int }_{M}Ric\left(\xi ,\xi \right)\ge {\int }_M\left(n(n-1){\sigma }^2{\alpha }^2+{∥F∥}^2\right).$$

(19)

Then, by regarding the hypersurface M and by considering a local frame $\left\{v_1,\dots ,v_n\right\}$, we can compute

$$\begin{array}{ccc}\hfill div\left(F\xi \right)& =& \sum _{i=1}^{n}g\left({\nabla }_{v_i}F\xi ,v_i\right)=\sum _{i=1}^{n}g\left(\left(\nabla F\right)\left(v_i,\xi \right)+F\left({\nabla }_{v_i}\xi \right),v_i\right)\hfill \\ & =& -\sum _{i=1}^{n}g\left(\xi ,\left(\nabla F\right)\left(v_i,v_i\right)\right)-\sum _{i=1}^{n}g\left({\nabla }_{v_i}\xi ,F{v}_i\right).\hfill \end{array}$$

By applying Equation (16), we can obtain

$$div\left(F\xi \right)=-\sum _{i=1}^{n}g\left(\xi ,\left(\nabla F\right)\left(v_i,v_i\right)\right)-\sum _{i=1}^{n}g\left(\sigma S{v}_i+F{v}_i,F{v}_i\right).$$

(20)

Note that, due to the symmetry property of S and the skew symmetry property of F, we ca obtain

$$\sum _{i=1}^{n}g\left(S{v}_i,F{v}_i\right)=0$$

(21)

Therefore, Equation (21) takes on the form

$$div\left(F\xi \right)=-\sum _{i=1}^{n}g\left(\xi ,\left(\nabla F\right)\left(v_i,v_i\right)\right)-{∥F∥}^2,$$

that is,

$$-\sum _{i=1}^{n}g\left(\xi ,\left(\nabla F\right)\left(v_i,v_i\right)\right)=div\left(F\xi \right)+{∥F∥}^2.$$

(22)

Next, we compute

$$\begin{array}{ccc}\hfill div\left(S\xi \right)& =& \sum _{i=1}^{n}g\left({\nabla }_{v_i}S\xi ,v_i\right)=\sum _{i=1}^{n}g\left(\left(\nabla S\right)\left(v_i,\xi \right)+S\left({\nabla }_{v_i}\xi \right),v_i\right)\hfill \\ & =& \sum _{i=1}^{n}g\left(\xi ,\left(\nabla S\right)\left(v_i,v_i\right)\right)+\sum _{i=1}^{n}g\left({\nabla }_{v_i}\xi ,S{v}_i\right).\hfill \end{array}$$

By utilizing Equations (15) and (16), we can obtain

$$div\left(S\xi \right)=g\left(\xi ,n\nabla \alpha \right)+\sum _{i=1}^{n}g\left(\sigma S{v}_i+F{v}_i,S{v}_i\right),$$

which, because of Equation (21), implies

$$div\left(S\xi \right)=n\xi \left(\alpha \right)+\sigma {∥S∥}^2.$$

(23)

Note that $div\left(\sigma S\xi \right)=\left(S\xi \right)\left(\sigma \right)+\sigma div\left(S\xi \right)$ and, consequently, Equation (23) imply

$$\left(S\xi \right)\left(\sigma \right)=div\left(\sigma S\xi \right)-n\sigma \xi \left(\alpha \right)-{\sigma }^2{∥S∥}^2.$$

(24)

Moreover, we have $n\alpha \xi \left(\sigma \right)=n\xi \left(\alpha \sigma \right)-n\sigma \xi \left(\alpha \right)$, that is,

$$-n\alpha \xi \left(\sigma \right)=n\sigma \xi \left(\alpha \right)-n\left(div\left(\sigma \alpha \xi \right)-\sigma \alpha div\left(\xi \right)\right).$$

However, Equation (16) gives us $div\left(\xi \right)=n\sigma \alpha$, and—by inserting this value in the above equation—we can thus obtain

$$-n\alpha \xi \left(\sigma \right)=n\sigma \xi \left(\alpha \right)+n^2{\sigma }^2{\alpha }^2-ndiv\left(\sigma \alpha \xi \right).$$

(25)

Inserting Equations (22), (24) and (25) into Equation (18) yields

$$Ric\left(\xi ,\xi \right)=div\left(\sigma S\xi \right)-{\sigma }^2{∥S∥}^2+n^2{\sigma }^2{\alpha }^2-ndiv\left(\sigma \alpha \xi \right)+div\left(F\xi \right)+{∥F∥}^2,$$

which, on integration, implies

$${\int }_{M}Ric\left(\xi ,\xi \right)={\int }_M\left(n^2{\sigma }^2{\alpha }^2-{\sigma }^2{∥S∥}^2+{∥F∥}^2\right).$$

Thus, we have

$${\int }_M{\sigma }^2\left({∥S∥}^2-n{\alpha }^2\right)={\int }_M\left(n(n-1){\sigma }^2{\alpha }^2+{∥F∥}^2\right)-{\int }_{M}Ric\left(\xi ,\xi \right).$$

In this equation, we use the inequality in (19), and we thus obtain the following:

$${\int }_M{\sigma }^2\left({∥S∥}^2-n{\alpha }^2\right)\le 0.$$

(26)

However, due to the Schwartz’s inequality, the integration on the left-hand side of the above inequality was found to be non-negative. Hence, we obtained

$${\sigma }^2\left({∥S∥}^2-n{\alpha }^2\right)=0.$$

Note that by using the connectedness of M and $\sigma \ne 0$, the above equation gives

$${∥S∥}^2=n{\alpha }^2,$$

which, with respect to the Schwartz’s inequality, is an equality that holds if and only if

$$S=\alpha I.$$

(27)

Thus, we can obtain

$$\left(\nabla S\right)\left(U,V\right)=U\left(\alpha \right)V,U,V\in \mathfrak{X}\left(M\right).$$

On the hypersurface M, when choosing a local orthonormal frame $\left\{v_1,\dots ,v_n\right\}$ and by considering the above equation, we can obtain

$$\sum _{i=1}^n\left(\nabla S\right)\left(v_i,v_i\right)=\sum _{i=1}^n{v}_i\left(\alpha \right)v_i=\nabla \alpha .$$

Using Equation (15) with the above equation, we can obtain

$$n\nabla \alpha =\nabla \alpha$$

Since $n>1$, we can infer that $\alpha$ is a constant. Consequently, when using Equation (27) in Equation (9), the curvature tensor of M is provided by

$$R\left(U,V\right)W={\alpha }^2\left\{g\left(V,W\right)U-g\left(U,W\right)V\right\},U,V,W\in \mathfrak{X}\left(M\right).$$

We can thus observe that, since M is a compact hypersurface of the Euclidean space ${\mathbf{R}}^{n+1}$, there exists a point on M where all sectional curvatures are of a positive nature. Consequently, we obtain the constant ${\alpha }^2>0$. As M is compact, it is thus complete. Therefore, M is complete and is a simply connected space of the constant positive curvature ${\alpha }^2$, thereby confirming that M is isometric to the sphere ${\mathbf{S}}^n\left({\alpha }^2\right)$.

Conversely, suppose M is isometric to the sphere ${\mathbf{S}}^n\left({\alpha }^2\right)$. By choosing a constant vector field $\overrightarrow{u}$ on the Euclidean space ${\mathbf{R}}^{n+1}$, and by defining a smooth function $\sigma$ on ${\mathbf{S}}^n\left({\alpha }^2\right)$ by $\sigma =⟨\overrightarrow{u},N⟩$ (where N is the unit normal to the sphere ${\mathbf{S}}^n\left({\alpha }^2\right)$), then let $\xi$ be the tangential projection of $\overrightarrow{u}$ to ${\mathbf{S}}^n\left({\alpha }^2\right)$. As such, we have

$$\overrightarrow{u}=\xi +\sigma N$$

(28)

together with the shape operator of the sphere ${\mathbf{S}}^n\left({\alpha }^2\right)$, which is given by $A=-\alpha I$. When differentiating Equation (28) and using the fundamental equation for a hypersurface, we obtain

$$0={\nabla }_U\xi -\alpha g\left(U,\xi \right)N+U\left(\sigma \right)N+\alpha \sigma U,U\in \mathfrak{X}\left({\mathbf{S}}^n\left({\alpha }^2\right)\right).$$

By equating the tangential and normal components, we can obtain

$${\nabla }_U\xi =-\sigma \alpha U,\nabla \sigma =\alpha \xi .$$

(29)

We can confirm

$$\frac{1}{2}\left({\mathcal{L}}_{\xi }g\right)\left(U,V\right)=-\sigma \alpha g\left(U,V\right)=\sigma g\left(AU,V\right),U,V\in \mathfrak{X}\left({\mathbf{S}}^n\left({\alpha }^2\right)\right),$$

where $\xi$ represents a shape vector field on ${\mathbf{S}}^n\left({\alpha }^2\right)$ with a potential function $\sigma$ and the associated operator $F=0$. Consequently, the Ricci curvature $Ric\left(\xi ,\xi \right)$ of the sphere ${\mathbf{S}}^n\left({\alpha }^2\right)$ is given by the following:

$$Ric\left(\xi ,\xi \right)=(n-1){\alpha }^2{∥\xi ∥}^2.$$

(30)

When using Equation (29), we obtain the Laplacian $\Delta \sigma =\alpha div\xi =-n{\alpha }^2\sigma$, that is, $\sigma \Delta \sigma =-n{\alpha }^2{\sigma }^2$. Integrating the last equation by parts yields

$${\int }_{{\mathbf{S}}^n\left({\alpha }^2\right)}{∥\nabla \sigma ∥}^2=n{\int }_{{\mathbf{S}}^n\left({\alpha }^2\right)}{\alpha }^2{\sigma }^2$$

(31)

When using $\nabla \sigma =\alpha \xi$ from Equation (29) in Equation (30), we can obtain

$${\int }_{{\mathbf{S}}^n\left({\alpha }^2\right)}Ric\left(\xi ,\xi \right)=(n-1){\int }_{{\mathbf{S}}^n\left({\alpha }^2\right)}{∥\nabla \sigma ∥}^2$$

In addition, by combining the above equation with Equation (31) and $F=0$, we can conclude

$${\int }_{M}Ric\left(\xi ,\xi \right)={\int }_M\left(n(n-1){\sigma }^2{\alpha }^2+{∥F∥}^2\right).$$

This completes the proof.

4. Proof of Theorem 2

Let M be a hypersurface that is compact and connected with a nonzero scalar curvature $\mathbf{ø}$ of ${\mathbf{R}}^{n+1}$, i.e., one that is associated with a unit normal N and the shape operator S, wherein the shape vector field $\xi$ with a potential function $\sigma$ is admitted. Suppose the function $\sigma \mathbf{ø}$ does not change the sign on M. From Equation (23), we can obtain

$$div\left(S\xi \right)=n\xi \left(\alpha \right)+\sigma {∥S∥}^2,$$

which, in view of $div\xi =n\sigma \alpha$ and $div\left(\alpha \xi \right)=\xi \left(\alpha \right)+n\sigma {\alpha }^2$ implies

$$div\left(S\xi \right)=ndiv\left(\alpha \xi \right)-n^2\sigma {\alpha }^2+\sigma {∥S∥}^2.$$

When using Equation (13), we can obtain

$$div\left(S\xi \right)=ndiv\left(\alpha \xi \right)-\sigma \mathbf{ø},$$

which, on integration, yields

$${\int }_M\sigma \mathbf{ø}=0.$$

Since the function $\sigma \mathbf{ø}$ does not change sign on M, the above equation implies $\sigma \mathbf{ø}=0.$ When connecting M with $\mathbf{ø}\ne 0$, we can obtain $\sigma =0$. Hence, Equation (1) implies

$${\mathcal{L}}_{\xi }g=0,$$

which, in turn, implies that $\xi$ is a Killing vector field.

5. Conclusions

In Theorem 1, we saw conditions under which the shape vector field $\xi$ is a conformal vector field. In Theorem 2, we obtained conditions under which the same will be a Killing vector field. Moreover, there are some important vector fields, namely Jacobi-type vector fields (cf. [11]) and concurrent vector fields (cf. [10]), on a Riemannian manifold that have interesting impacts on the geometry of the host manifold. Finding the conditions under which a shape vector field on hypersurface of a Euclidean apace $R^{n+1}$ could be reduced to a Jacobi-type vector field or to a concurrent vector field could present an thought-provoking question.

The geodesic vector field (cf. [22]) is also considered another important special vector field on a Riemannian manifold. On a Riemannian manifold $(M,g)$, a vector field $\zeta$ is said to be a geodesic vector field if the integral curves of $\zeta$ are geodesics. Naturally, a Killing vector field has this property locally (i.e., that its integral curves are geodesics in a neighborhood), but this cannot be extended globally. There are numerous examples of geodesic vector fields that are not Killing vector fields, and these examples are provided by non-Sasakian structures such as Kenmotsu manifolds and nontrivial trans-Sasakian manifolds (cf. [23]). Therefore, it will be of interest to find the conditions on the shape vector field on a hypersurface M of the Euclidean space $R^{n+1}$, such that it is reduced to a geodesic vector field. It is worth mentioning that we could analyze the applicational aspect of a shape vector field on the hypersurface of spacetime spaces, and it is also the case that the content in the books of (cf. [24,25]) would be helpful for this purpose.