A Note on Shape Vector Fields on Hypersurfaces

: 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.


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 α, scalar curvature τ, 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 is necessarily isometric to the n-sphere S n α 2 .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 ξ on the hypersurface M, namely that ξ is the tangential component of the position vector field T, which is required to obtain the following integral formula where ρ 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 ξ, such that the function g(Sξ, ξ) 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 → 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 ξ-which is defined as a vector field for which the local flow corresponds to conformal transformations of (M, g)-is equivalently the following: where the Lie derivative of the metric g with respect to the vector field ξ is represented as L ξ g, while σ denotes a smooth function on M (which is referred to as the conformal factor associated with ξ).If σ = 0, then ξ is said to be a Killing vector field.For M, an orientable hypersurface of the Riemannian manifold (M, g) is equipped with a shape operator S and a unit normal N.If we define a smooth vector field ξ on the hypersurface M, it is called a shape vector field if it meets the following conditions: where B is defined on M by and the space of vector fields on the hypersurface M is denoted as X(M).The potential function of the shape vector field ξ defined on the hypersurface M is denoted by the smooth function σ.In the case of the unit sphere S 2n+1 , as a hypersurface of Euclidean space R 2(n+1) , the Reeb vector field ξ on S 2n+1 is a shape vector field with a potential function σ = 0. We will see that there are shape vector fields on the sphere S n (c), which have a constant curvature c as the hypersurfaces of the Euclidean space R n+1 (which also have a nonzero potential functions, as shown in Section 3).Consider an orientable hypersurface M in R n+1 that is characterized by a unit normal vector N and a shape operator S. Let ξ be a shape vector field on M with a corresponding potential function σ.We will use η to denote the 1-form that is dual to the shape vector field ξ, that is, where g is the induced metric on M.Then, the operator F is defined on the hypersurface such that dη is the exterior derivative of η.We call F the operator associated with the shape vector field.For a local orthonormal frame {v 1 , . . . ,v n }, which is defined on the hypersurface, the squared length of F is represented as We denote the above by α as the mean curvature of the hypersurface M, and this is defined by 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 S n (c).
Theorem 1.Let M be compact and simply connected to the hypersurface of R n+1 , n ≥ 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(ξ, ξ) satisfies if and only if α is a constant and M is isometric to the sphere S n α 2 .
In the methods of the proof of Theorem 1, we found that, due to the restriction on the Ricci curvature, the shape vector field ξ 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 ø of R n+1 , and let ξ be a shape vector field with potential function σ on M. If the function σø 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 τ of the hypersurface, which is a constant and has a Ricci curvature that is given by The Fischer-Marsden equation is considered one of the most significant differential equations on a Riemannian manifold (M, g), and it is given by (∆ρ)g + ρRic = Hess(g) 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 τ 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 ρ 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 ρ of the FM-equation is admitted.Then, by undertaking a trace in Equation ( 4), we can obtain By taking ξ = ∇ρ on the hypersurface M, i.e., when using Equation (4), we can conclude Since M is non-total geodesic completely umbilical hypersurface, we can obtain SU = αU, the constant α = 0 and-consequently-Equation (6), which implies that Hence, on the hypersurface M of the Euclidean space R n+1 , ξ is a shape vector field.

Preliminaries
For an orientable hypersurface M of the Euclidean space 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 ∇ and ∇ by the Euclidean connection and the Riemannian connection on R n+1 and M, respectively.Then, for the hypersurface M, the fundamental equations are as follows: and For the hypersurface M, the curvature tensor R is given by For the hypersurface M, the Ricci tensor is given by where the hypersurface M and the mean curvature α is denoted by nα = Tr • S. In addition, the Ricci operator Q is defined by and the scalar curvature ø is introduced by Using Equation (10), we can obtain where and {v 1 , . . . ,v n } is a local frame on the hypersurface M. As the Euclidean space R n+1 is flat, the Codazzi equation for the hypersurface has the form where Also, by considering a local frame {v 1 , . . . ,v n } on the hypersurface M, and by using nα = Tr • S, i.e., and then using the symmetry of the shape operator S, as well as Equation ( 14), we can compute Thus, the mean curvature α has the gradient Regarding the hypersurface M, consider ξ as the shape vector field with the potential function σ and the associated operator F.Then, we can obtain where B(U, V) = g(SU, V) and η are the 1-form that is dual to ξ.Then, by applying Koszul's formula (cf.[10]), we can obtain When using the formula of the curvature tensor and Equations ( 14) and ( 16), we can confirm Regarding the hypersurface M, when using a local frame {v 1 , . . . ,v n } and the following formula of the Ricci tensor as well as Equation ( 17), we arrive at In this context, the symmetry of the shape operator S and the equation nα = Tr(S) have been utilized.Now, by applying the skew symmetry property of the operator F, its outcomes Tr.F = 0 and we can conclude that is,

Proof of Theorem 1
Regarding M, there is a hypersurface of R n+1 , n ≥ 2, which is compact and simply connected with a unit normal vector N and a shape operator S, where a shape vector field ξ that has a nonzero potential function σ and a related operator Then, by regarding the hypersurface M and by considering a local frame {v 1 , . . . ,v n }, we can compute By applying Equation ( 16), we can obtain Note that, due to the symmetry property of S and the skew symmetry property of F, we ca obtain Therefore, Equation ( 21) takes on the form Next, we compute By utilizing Equations ( 15) and ( 16), we can obtain which, because of Equation ( 21), implies Note that div(σSξ) = (Sξ)(σ) + σdiv(Sξ) and, consequently, Equation (23) imply Moreover, we have nαξ(σ) = nξ(ασ) − nσξ(α), that is, −nαξ(σ) = nσξ(α) − n(div(σαξ) − σαdiv(ξ)).
However, Equation (16) gives us div(ξ) = nσα, and-by inserting this value in the above equation-we can thus obtain Inserting Equations ( 22), ( 24) and (25) into Equation (18) yields which, on integration, implies Thus, we have In this equation, we use the inequality in (19), and we thus obtain the following: 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 Note that by using the connectedness of M and σ = 0, the above equation gives which, with respect to the Schwartz's inequality, is an equality that holds if and only if Thus, we can obtain (∇S)(U, V) = U(α)V, U, V ∈ X(M).
On the hypersurface M, when choosing a local orthonormal frame {v 1 , . . . ,v n } and by con- sidering the above equation, we can obtain Using Equation (15) with the above equation, we can obtain n∇α = ∇α Since n > 1, we can infer that α is a constant.Consequently, when using Equation (27) in Equation ( 9), the curvature tensor of M is provided by We can thus observe that, since M is a compact hypersurface of the Euclidean space R n+1 , there exists a point on M where all sectional curvatures are of a positive nature.Consequently, we obtain the constant α 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 α 2 , thereby confirming that M is isometric to the sphere S n α 2 .
Conversely, suppose M is isometric to the sphere S n α 2 .By choosing a constant vector field − → u on the Euclidean space R n+1 , and by defining a smooth function σ on M σø = 0.
Since the function σø does not change sign on M, the above equation implies σø = 0.When connecting M with ø = 0, we can obtain σ = 0. Hence, Equation (1) implies L ξ g = 0, which, in turn, implies that ξ is a Killing vector field.

Conclusions
In Theorem 1, we saw conditions under which the shape vector field ξ 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 ζ is said to be a geodesic vector field if the integral curves of ζ 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.