In 1903 H. Minkowski published in [16] his two famous integral formulas for compact surfaces in three dimensional Euclidean space. After that, many authors obtained integral formulas that generalized the two Minkowski formulas to hypersurfaces in Euclidean space and then in a general Riemannian manifold that admits a Killing or conformal vector field. For instance, in [11] and [12], C. C. Hsiung obtained generalized integral formulas of Minkowsi type for embedded hypersurfaces in Riemannian manifolds (see also [13]). In [14] and [15], Y. Katsurada generalized the work of Hsiung and derived some integral formulas of Minkowski type that are valid for Einstein manifolds and used them to prove that given a hypersurface $\left(M,g\right)$ with constant mean curvature in an Einstein Riemannian manifold $\left(\overline{M},\overline{g}\right)$, and given a homothetic vector field $\xi$ of $\left(\overline{M},\overline{g}\right)$ such that the inner product of $\xi$ and the normal to M does not change sign and does not vanish on M, then M is necessarily umbilical. In [19], K. Yano obtained three integral formulas of Minkowski type for hypersurfaces with constant mean curvature in a Riemannian manifold admitting a homothetic vector field. Then, over time, several integral formulas of Minkowski type appeared in literature that were used to obtain rigidity results for isometrically immersed hypersurfaces in pseudo-Riemannian manifolds admitting a conformal vector filed. In [6] and [7] (resp. [8]), L. J. Alias, A. Romero, and M. Sanchez obtained the first and second integral formulas of Minkowski type for compact spacelike hypersurfaces in a generalized Robertson–Walker spacetime (resp. conformally stationary spacetime), and applied them to the study of compact spacelike hypersurfaces with constant mean curvature. Two years later, in [17], S. Montiel provided another proof of the first and second Minkowski formulas in the case where the ambient spacetime is equipped with a conformal timelike vector field. In 2003, L. J. Alias, A. Brasil JR, and A. G. Colares generalized in [2] the integral formulas obtained in [6,7,8] for spacelike hypersurfaces in conformally stationary spacetimes. See also [3] and [4].

The assumption that the ambient space admits a conformal vector field is inspired by the fact that the position vector field in Euclidean space is a closed conformal vector field (which in some references is called a concircular vector field). The importance of conforml vector fields comes from the use of conformal mappings as a mathematical tool in general relativity. In fact, although a conformal vector field does not leave the Einstein tensor invariant, its existence in a pseudo-Riemannian manifold $\left(M,g\right)$ is a symmetry assumption for g that can be used (for example) to obtain exact solutions of Einstein’s equation.

Consider now an $\left(n+1\right)$-dimensional either Riemannian or Lorentzian manifold $\left(\overline{M},\overline{g}\right)$ admitting a conformal vector field $\overline{\xi }$ that we assume to be timelike in the case where $\left(\overline{M},\overline{g}\right)$ is Lorentzian. Let $\left(M,g\right)$ be a connected $n$-dimensional Riemannian manifold that is isometrically immersed as a hypersurface into $\left(\overline{M},\overline{g}\right)$, and let $\xi$ denote the restriction of $\overline{\xi }$ to $M.$ Consider the function $\theta =\overline{g}\left(\xi ,N\right)$, where $\overline{\xi }$ is an arbitrary vector field and N is a globally defined unit vector field normal to $M.$ In the case where is Riemannian, L. J. Alias, M. Dajczer, and J. Ripoll gave in [5] an expression for the Laplacian $\Delta \theta$ in terms of the Ricci curvature of $\left(\overline{M},\overline{g}\right)$ and the norm of the shape operator of $\left(M,g\right)$. One year later, in 2008, A. Barros, A. Brasil, and A. Caminha obtained in [9] the analogous expression when $\left(\overline{M},\overline{g}\right)$ is Lorentzian.

In 2010, A. L. Albujer, J. A. Aledo, and L. J. Alias gave in [1] an expression for $\Delta \theta$ in a slightly different way as given in [5] and [9]. Then, they used this expression to obtain a Minkowski-type integral formula for compact Riemannian and spacelike hypersurfaces, and applied this to deduce some interesting results concerning the characterization of compact Riemannian and spacelike hypersurfaces under certain hypotheses like the constancy of the mean curvature of the assumption that the ambient space is Einstein or a product space.

In this paper, we mainly wish to generalize previous results concerning Minkowski-type integral formulas for Riemannian (resp. spacelike) hypersurfaces in Riemannian (resp. Lorentzian) manifolds admitting an arbitrary vector field that we assume to be timelike in the case where is Lorentzian, and apply these integral forms to compact Riemannian and spacelike hypersurfaces in order to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases, such as the ambient space is Einstein admitting an arbitrary (and in particular, a conformal Killing) vector field, or the hypersurface is minimal (resp. maximal) or has constant mean curvature.

In particular, we wish to work to generalize the results in [19] and [1] for any arbitrary Riemannian or spacelike hypersurface in any arbitrary ambient space admitting an arbitrary vector field. More precisely, given an $\left(n+1\right)$-dimensional either Riemannian or Lorentzian manifold $\left(\overline{M},\overline{g}\right)$ admitting an arbitrary vector field $\overline{\xi }$ that we assume to be timelike in the case where $\left(\overline{M},\overline{g}\right)$ is Lorentzian, and given a connected $n$-dimensional Riemannian manifold $\left(M,g\right)$ that is isometrically immersed as a hypersurface into $\left(\overline{M},\overline{g}\right)$. Let $\xi$ denote the restriction of $\overline{\xi }$ to M, and let N be a globally defined unit vector field normal to $M.$ Of course, N is supposed to be timelike in the case where $\left(\overline{M},\overline{g}\right)$ is Lorentzian. Our first main goal in this paper is to give a useful expression for the Laplacian $\Delta \theta$ of the function $\theta =\overline{g}\left(\xi ,N\right)$ in terms of the Ricci and scalar curvatures of the ambient space, the mean curvature of the hypersurface, and the tangent part of the restriction of the vector field $\xi$ to M. In the particular case where $\overline{\xi }$ is a conformal (resp. Killing) vector field, our expression reduces to that obtained in [1] (resp. [19]). We will deduce from the generalized expression for $\Delta \theta$ different generalized Minkowski-type integral formulas valid for any Riemannian or spacelike hypersurface in any arbitrary Riemannian or Lorentzian manifold admitting an arbitrary vector field. We will, in particular, generalize an integral formula obtained in [1] in the case when $\overline{\xi }$ is conformal to the case of an arbitrary vector field. We will also apply the obtained generalized Minkowski-type formulas to deduce interesting results concerning the characterization of Riemannian and spacelike hypersurfaces in some particular cases, such as the ambient space is Einstein admitting an arbitrary (and in particular, a conformal Killing) vector field, or the hypersurface has constant mean curvature.

Let $n\ge 2,$ and let $\left(M,g\right)$ be a connected $n$-dimensional pseudo-Riemannian manifold. In this paper, we adopt the opposite convention of that in [18] to define the Riemannian tensor. That is, the Riemannian tensor is defined here to be the $\left(1,3\right)$ tensor field given by for all $X,Y,Z\in \mathfrak{X}\left(M\right)$.

For every $p\in M$ and every orthonormal basis $\left\{e_1,\dots ,e_n\right\}$ of $T_{p}M$, the Ricci curvature tensor $Ric$ and the scalar curvature $Scal$ are, respectively, defined to be for all $X,Y\in T_{p}M.$ where ${ϵ}_i=g\left(e_i,e_i\right).$

Throughout this paper, we will assume that $\left(M,g\right)$ is Riemannian (i.e. the metric g has index 0) which is isometrically immersed as a hypersurface into an $\left(n+1\right)$-dimensional pseudo-Riemannian manifold $\left(\overline{M},\overline{g}\right)$ that we assume to be Riemannian or Lorentzian (i.e. the metric $\overline{g}$ has index 0 or 1). Let ∇ and $\overline{\nabla }$ denote the Levi-Civita connections on M and $\overline{M}$, respectively. Let $\mathfrak{X}\left(M\right)$ and $\mathfrak{X}\left(\overline{M}\right)$ denote, respectively, the sets of all tangent vector fields on M and $\overline{M,}$ and let $\overline{\mathfrak{X}}\left(M\right)$ denote the set of all $\overline{M}$ vector fields on $M.$ We will use the two notations $X·f$ or $X\left(f\right)$ to denote the value of a vector field X on a function f.

Let $\overline{\xi }\in \mathfrak{X}\left(\overline{M}\right)$ that we will assume to be timelike in the case where $\overline{M}$ is Lorentzian, and let ${\theta }_{\overline{\xi }}$ denote its dual one form, that is, the one form given by ${\theta }_{\overline{\xi }}\left(Y\right)=\overline{g}\left(\overline{\xi },Y\right)$, for every $Y\in \mathfrak{X}\left(\overline{M}\right).$ Let $A_{\overline{\xi }}$ be the $\left(1,1\right)$-tensor (viewed as an endomorphism) defined by

Write as usual for all $Y,Z\in \mathfrak{X}\left(\overline{M}\right),$ where L is the Lie derivative of the metric $\overline{g}$ with respect to $\overline{\xi }$ a

Let B and $\varphi$ be the symmetric and skew-symmetric parts of $A_{\overline{\xi }}.$ In other words, we have

Now, in the case where $\overline{M}$ is Riemannian, we will assume that there exists a globally defined unit vector field N normal to $M.$ In this case, M is said to be a two-sided hypersurface. In the case where $\overline{M}$ is Lorentzian, since M is a spacelike hypersurface in $\overline{M}$ and $\overline{\xi }$ is assumed to be timelike, then we can choose a (globally defined) timelike unit vector field N normal to M and in the same time-orientation of $\overline{\xi }$, that is we have $\overline{g}\left(\overline{\xi },N\right)<0$ on $M.$ In both cases, if $\xi$ is the restriction of $\overline{\xi }$ to M, then we will denote by $\theta$ the smooth function on M, called the support function that is defined by $\theta =\overline{g}\left(\xi ,N\right)$. It is clear that in the case where $\overline{M}$ is Lorentzian we have $\theta \le -\sqrt{-\overline{g}\left(\xi ,\xi \right)}<0$. If T is the tangential component of $\xi$ to $M,$then we have where $ϵ=\overline{g}\left(N,N\right)=\pm 1,$ according to whether $\overline{M}$ is Riemannian or Lorentzian, respectively. Since $\xi \in \overline{\mathfrak{X}}\left(M\right)$, then the operator $A_{\xi }:\mathfrak{X}\left(M\right)\to \overline{\mathfrak{X}}\left(M\right)$ given by $A_{\xi }\left(X\right)={\overline{\nabla }}_X\xi$ is well-defined (see for instance [18], pp. 97-99). Then, we have where $\psi \left(X\right)={\left({\overline{\nabla }}_X\xi \right)}^{\top }$ is the tangential component of ${\overline{\nabla }}_X\xi$ to M and $\alpha$ is a one form on $M.$ Let $\eta \in \mathfrak{X}\left(M\right)$ be the vector field associated to $\alpha .$ Therefore, for all $X\in \mathfrak{X}\left(M\right),$ we have

Since $\varphi$ is skew-symmetric, we have $\overline{g}\left(\varphi \left(N\right),N\right)=0$, that is $\varphi \left(N\right)\in \mathfrak{X}\left(M\right)$. Therefore, (5) implies that

On the other hand, the Gauss and Weingarten formulae for M as a hypersurface of $\overline{M}$ are given by for all $X,Y\in \mathfrak{X}\left(M\right),$ where A is the shape operator of M with respect to $N.$ Therefore, for all $X\in \mathfrak{X}\left(M\right)$, we have

From (4) and (7) we deduce that

For that we need to recall some definitions. In general, recall that for a $\left(1,1\right)$-tensor S, the covariant derivative $\nabla S$ of S is defined as follows

The divergence of a vector field $X\in \mathfrak{X}\left(M\right)$ is defined as the function where $\left\{e_1,\dots ,e_n\right\}$ is a local orthonormal frame of vector fields in $\mathfrak{X}\left(M\right).$

The divergence of a $\left(1,1\right)$-tensor S on M is defined as the vector field where, as above, $\left\{e_1,\dots ,e_n\right\}$ is a local orthonormal frame of vector fields in $\mathfrak{X}\left(M\right)$.

We observe that, without loss of generality, we may assume $\left\{e_1,\dots ,e_n\right\}$ to be parallel. In this case, we see that

We also recall that the curvature tensor R of M is given in terms of the curvature tensor $\overline{R}$ of $\overline{M}$ and the shape operator by the so-called Gauss equation for all $X,Y,Z\in \mathfrak{X}\left(M\right)$.

Recalling that the mean curvature of M is defined to be it follows from (11) that the Ricci curvatures $Ric$ and $\overline{Ric}$ of M and $\overline{M}$ are related as follows for all $X,Y\in \mathfrak{X}\left(M\right)$.

Also, by tracing (13), we see that the scalar curvatures $Scal$ and $\overline{Scal}$ of M and $\overline{M}$ are related as follows

With the notations above, let $\left(\overline{M},\overline{g}\right)$ be an $\left(n+1\right)$-dimensional either Riemannian or Lorentzian manifold, and let $\overline{\xi }\in \mathfrak{X}\left(\overline{M}\right)$ be an arbitrary vector field that we assume to be timelike in the case where $\left(\overline{M},\overline{g}\right)$ is Lorentzian. Let $\left(M,g\right)$ be a connected $n$-dimensional Riemannian manifold that is isometrically immersed as a hypersurface into $\left(\overline{M},\overline{g}\right)$, and let $\xi$ denote the restriction of $\overline{\xi }$ to $M.$

Our main goal in this section is to give a useful expression for the Laplacian $\Delta \theta$ of the function $\theta =\overline{g}\left(\xi ,N\right)$, where $\overline{\xi }$ is an arbitrary vector field and N is a globally defined unit vector field normal to $M.$ In the case where $\left(\overline{M},\overline{g}\right)$ is Riemannian and $\overline{\xi }$ is a Killing (resp. conformal) vector field an expression for $\Delta \theta$ has been given in [10] (resp. [5]) in terms of the Ricci curvature of $\left(\overline{M},\overline{g}\right)$ and the norm of the shape operator. An analogous formula has been obtained in [9] in the case where $\left(\overline{M},\overline{g}\right)$ is Lorentzian and $\overline{\xi }$ is a timelike conformal vector field. As we have mentioned in the introduction, in [1], a formula for $\Delta \theta$ was obtained in a slightly different way as given in [5] and [9].

Let us denote by $B^{\top }$ the restriction of B to $TM,$ and let $f=\overline{g}\left({\overline{\nabla }}_N\xi ,N\right)$. It is clear that f is a smooth function on $M.$ In fact, from (1), we see that

To calculate $\Delta \theta ,$ we will use (). So, we start by computing the divergences of T and $A\left(T\right).$

In the following proposition, we give an explicit useful formula for $div\left(A\left(T\right)\right)$ in terms of the Ricci curvature (compare to formula (14) in [1]).

Now, from this last expression and (17) we obtain (18). We now give an expression for $div{\left({\overline{\nabla }}_N\xi \right)}^{\top }.$ For this purpose, we note that from which we have

Since ${\overline{\nabla }}_N\xi =B\left(N\right)+\varphi \left(N\right)$ and $\varphi \left(N\right)\in \mathfrak{X}\left(M\right),$ it follows that

We are now ready to give the desired expression for $\Delta \theta .$

As a straightforward consequence of Theorem 4, we obtain the interesting expression for $\Delta \theta$ in the particular case where $\overline{\xi }$ is a conformal Killing vector field on $\overline{M}$, that is a vector field satisfying for some $\psi$ smooth function on $\overline{M},$ called the conformal factor (or potential function) of $\overline{\xi }.$

It would be of some use to express $\Delta \theta$ in terms of the scalar curvatures of M and $\overline{M}$. This can be done by combining the two formulas (14) and (26), so that we get formula (31) in the following theorem.

It should be noticed that formula (31) is nothing but a generalization to the case of an arbitrary vector field on $\overline{M}$ of formula (9) in [1] which was given in the case where $\overline{\xi }$ is a conformal Killing vector field.

On the other hand, we give an expression for $div\left({\left({\overline{\nabla }}_N\xi \right)}^{\perp }\right).$

In this section, we assume that $\left(M,g\right)$ is an n-dimensional compact Riemannian manifold that is isometrically immersed as a hypersurface in an $\left(n+1\right)$-dimensional either Riemannian or Lorentzian manifold $\left(\overline{M},\overline{g}\right)$ with all the assumptions stated at the beginning of the above section. The first integral formula that we can display here results directly from the integration of the simple formula (16).

By using formula (42) of the previous proposition, the following results can be easily deduced.

As an immediate consequence of Proposition 10, we have the following

A more general result than Proposition 10 is the following

Our second integral formula is the following

Our third integral formula is the following

In this section, we will focus on the case when M has constant mean curvature $H.$The first result gives an integral formula for a hypersurface with constant mean curvature without any assumption on the ambient space $\left(\overline{M},\overline{g}\right)$ or on the vector field $\overline{\xi }.$