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.
Keywords:
shape vector fields; conformal vector fields; Killing vector fields; Euclidean space; Ricci curvature; spheres MSC:
35C08; 35Q51; 53B05
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 , scalar curvature , the squared norms of the shape operator S, the Ricci operator Q and the curvature tensor field R. In [], it was shown that a compact hypersurface M of the Euclidean space satisfying the inequality
is necessarily isometric to the n-sphere . In [], the position vector field T of a compact hypersurface M in the Euclidean space was used to define a smooth vector field on the hypersurface M, namely that is the tangential component of the position vector field T, which is required to obtain the following integral formula
where 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 (cf. []). 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 [], it was shown that a compact hypersurface M of the Euclidean space admits a Killing vector field unit , such that the function 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 , namely the translation hypersurface. In this type of hypersurface, the focus is on the analysis of the smooth function , whose graph is the desired hypersurface under study. For instance, in [], the author investigates translation hypersurfaces in the Euclidean space , 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 . In [], 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 has also been introduced (cf. [,,,]). Regarding separable hypersurfaces, the defining smooth function, whose graph is the translation hypersurface, is required to satisfy the additional conditions. In [], the authors studied separable hypersurfaces in , 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. [,,,,,,,,,]). On a Riemannian manifold , a conformal vector field —which is defined as a vector field for which the local flow corresponds to conformal transformations of —is equivalently the following:
where the Lie derivative of the metric g with respect to the vector field is represented as , while denotes a smooth function on M (which is referred to as the conformal factor associated with ). If , then is said to be a Killing vector field. For M, an orientable hypersurface of the Riemannian manifold is equipped with a shape operator S and a unit normal N. If we define a smooth vector field 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 . The potential function of the shape vector field defined on the hypersurface M is denoted by the smooth function . In the case of the unit sphere , as a hypersurface of Euclidean space , the Reeb vector field on is a shape vector field with a potential function . We will see that there are shape vector fields on the sphere , which have a constant curvature c as the hypersurfaces of the Euclidean space (which also have a nonzero potential functions, as shown in Section 3).
Consider an orientable hypersurface M in that is characterized by a unit normal vector N and a shape operator S. Let be a shape vector field on M with a corresponding potential function . We will use to denote the 1-form that is dual to the shape vector field , that is,
where g is the induced metric on M. Then, the operator F is defined on the hypersurface M by
such that is the exterior derivative of . We call F the operator associated with the shape vector field. For a local orthonormal frame , which is defined on the hypersurface, the squared length of F is represented as
We denote the above by as the mean curvature of the hypersurface M, and this is defined by
where 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 .
Theorem 1.
Let M be compact and simply connected to the hypersurface of , , 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 satisfies
if and only if α is a constant and M is isometric to the sphere .
In the methods of the proof of Theorem 1, we found that, due to the restriction on the Ricci curvature, the shape vector field 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 of , and let ξ be a shape vector field with potential function σ on M. If the function 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 . The geometry of Einstein hypersurfaces in a Euclidean space is very rich (cf. [,]). Thus, let M be a non-total geodesic that is a completely umbilical Einstein hypersurface of the Euclidean space . Then, we will obtain the scalar curvature 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 , and it is given by
We shall call the above equation the FM-equation for convenience (cf. []). It has been widely acknowledged that, in a Riemannian manifold , the scalar curvature admits a nontrivial solution of FM-equation that is a constant (cf. []). Indeed, it has been conjectured that a compact has a nontrivial solution for the FM-Equation (4), which should imply that 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. [,,]). As such, let M be a non-total geodesic that is a completely umbilical Einstein hypersurface of the Euclidean space , wherein a nontrivial solution of the FM-equation is admitted. Then, by undertaking a trace in Equation (4), we can obtain
By taking 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 , the constant and—consequently—Equation (6), which implies that
that is,
Hence, on the hypersurface M of the Euclidean space , is a shape vector field.
2. Preliminaries
For an orientable hypersurface M of the Euclidean space (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 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 is denoted by . In addition, the Ricci operator Q is defined by
and the scalar curvature is introduced by
Using Equation (10), we can obtain
where
and is a local frame on the hypersurface M. As the Euclidean space is flat, the Codazzi equation for the hypersurface has the form
where
Also, by considering a local frame on the hypersurface M, and by using , i.e.,
and then using the symmetry of the shape operator S, as well as Equation (14), we can compute
Thus, the mean curvature has the gradient
Regarding the hypersurface M, consider as the shape vector field with the potential function and the associated operator F. Then, we can obtain
where and are the 1-form that is dual to . Then, by applying Koszul’s formula (cf. []), 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 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 have been utilized. Now, by applying the skew symmetry property of the operator F, its outcomes and
we can conclude
that is,
3. Proof of Theorem 1
Regarding M, there is a hypersurface of , , which is compact and simply connected with a unit normal vector N and a shape operator S, where a shape vector field that has a nonzero potential function and a related operator F is admitted. Suppose the Ricci curvature of hypersurface M satisfies
Then, by regarding the hypersurface M and by considering a local frame , 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
that is,
Next, we compute
By utilizing Equations (15) and (16), we can obtain
which, because of Equation (21), implies
Note that and, consequently, Equation (23) imply
Moreover, we have , that is,
However, Equation (16) gives us , 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 , 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
On the hypersurface M, when choosing a local orthonormal frame and by considering the above equation, we can obtain
Using Equation (15) with the above equation, we can obtain
Since , we can infer that 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 , there exists a point on M where all sectional curvatures are of a positive nature. Consequently, we obtain the constant . As M is compact, it is thus complete. Therefore, M is complete and is a simply connected space of the constant positive curvature , thereby confirming that M is isometric to the sphere .
Conversely, suppose M is isometric to the sphere . By choosing a constant vector field on the Euclidean space , and by defining a smooth function on by (where N is the unit normal to the sphere ), then let be the tangential projection of to . As such, we have
together with the shape operator of the sphere , which is given by . When differentiating Equation (28) and using the fundamental equation for a hypersurface, we obtain
By equating the tangential and normal components, we can obtain
We can confirm
where represents a shape vector field on with a potential function and the associated operator . Consequently, the Ricci curvature of the sphere is given by the following:
When using Equation (29), we obtain the Laplacian , that is, . Integrating the last equation by parts yields
When using from Equation (29) in Equation (30), we can obtain
In addition, by combining the above equation with Equation (31) and , we can conclude
This completes the proof.
4. Proof of Theorem 2
Let M be a hypersurface that is compact and connected with a nonzero scalar curvature of , i.e., one that is associated with a unit normal N and the shape operator S, wherein the shape vector field with a potential function is admitted. Suppose the function does not change the sign on M. From Equation (23), we can obtain
which, in view of and implies
When using Equation (13), we can obtain
which, on integration, yields
Since the function does not change sign on M, the above equation implies When connecting M with , we can obtain . Hence, Equation (1) implies
which, in turn, implies that is a Killing vector field.
5. Conclusions
In Theorem 1, we saw conditions under which the shape vector field 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. []) and concurrent vector fields (cf. []), 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 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. []) is also considered another important special vector field on a Riemannian manifold. On a Riemannian manifold , a vector field is said to be a geodesic vector field if the integral curves of 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. []). Therefore, it will be of interest to find the conditions on the shape vector field on a hypersurface M of the Euclidean space , 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. [,]) would be helpful for this purpose.
Funding
This research received no external funding.
Data Availability Statement
Data are contained within the article.
Acknowledgments
The author would like to thank the Arab Open University for supporting this work.
Conflicts of Interest
The author declares no conflict of interest.
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