The Reilly ’ s Integral Formula on Semi-Riemannian Manifolds with Nondegenerate Boundary

In Riemannian geometry, integral formulas have been studied by many mathematicians [1] and their use many beatiful global results have been obtained. Perhaps the Reilly’s formula is one of the most well known integral formula in Riemannian geometry as well as a very powerful tool for obtaining global results. Nonetheless, a Reilly’s Formula in semi-Riemannian geometry has not been available. The main difficulty in stating an integral formula for semi-Riemannian manifolds is that the boundary may become degenerate at some points and hence there exists no well-defined unit outward normal at such points. Consequently there is no well defined induced volume form on the boundary. Duggal was the first one who studied semi-Riemannian manifolds with boundary in one of his works on integral formulas in semi-Riemannian geometry [2]. In [2], Duggal defined a semi-Riemannian manifold to be regular if the usual form of integral formulas remains valid on it. In [3], Ünal defined nondegenerate boundary of a semi-Riemannian manifold and by making use of the volume form on the nondegenerate boundary, he obtained integral formulas. In this paper, we define two type semi-Riemannian inner product. Using this definition we classify the boundaries.We define nondegenerate boundary of a semiRiemannian manifold and we get Reilly’s formula on the nondegenerate boundary. Of course, the validity of the Reilly’s formula depends on some restrictions, namely, the degenerate part of the boundary must have measure zero. Finally, we obtain different results from Riemannian geometry.


Introduction
In Riemannian geometry, integral formulas have been studied by many mathematicians [1] and their use many beatiful global results have been obtained.Perhaps the Reilly's formula is one of the most well known integral formula in Riemannian geometry as well as a very powerful tool for obtaining global results.Nonetheless, a Reilly's Formula in semi-Riemannian geometry has not been available.The main difficulty in stating an integral formula for semi-Riemannian manifolds is that the boundary may become degenerate at some points and hence there exists no well-defined unit outward normal at such points.Consequently there is no well defined induced volume form on the boundary.
Duggal was the first one who studied semi-Riemannian manifolds with boundary in one of his works on integral formulas in semi-Riemannian geometry [2].In [2], Duggal defined a semi-Riemannian manifold to be regular if the usual form of integral formulas remains valid on it.In [3], Ünal defined nondegenerate boundary of a semi-Riemannian manifold and by making use of the volume form on the nondegenerate boundary, he obtained integral formulas.
In this paper, we define two type semi-Riemannian inner product.Using this definition we classify the boundaries.We define nondegenerate boundary of a semi-Riemannian manifold and we get Reilly's formula on the nondegenerate boundary.Of course, the validity of the Reilly's formula depends on some restrictions, namely, the degenerate part of the boundary must have measure zero.Finally, we obtain different results from Riemannian geometry.

Basic Notions and Terminologies
Let R n be n-dimensional real vector space.Semi-Riemannian inner product for n-dimensional real vector space R n is defined as follows [4]: x j y j or x j y j In addition, β i and ε i are defined as follows: and Considering β i and ε i in Eq. (2.1) and Eq.(2.2), respectively, we get Here the functions of , 1 and , 2 are semi-Riemannian inner product in R n and R n v is semi-Riemannian space which is united with the functions of , 1 and , 2 .
For the sake of shortness, let's unite both of semi-Riemannian inner product definition and let's express this definition as follows: If is written as the following, then semi-Riemannian inner product is as follows: The Reilly's Integral Formula on Semi-Riemannian Manifolds 127 Throughout this paper, let M denote an n-dimensional semi-Riemannian manifold with metric , of index 0 ≤ v ≤ n and boundary ∂M .Then the open submanifold ∂M ′ = ∂M + ∪ ∂M − of ∂M is called the nondegenerate boundary of (M, , ).A vector 0 = v ∈ T M is respectively called spacelike, timelike and null if , > 0, , < 0, , = 0. We will also assume that M is oriented and ∂M is oriented by the induced orientation.Also let dv be the semi-Riemannian volume element on M, that is, dv is an exterior n-form on M with for semi-Riemannian orthonormal basis [5] and let e n be the unit outward normal vector field on the nondegenerate boundary ∂M ′ of (M, , ).Let M be an n-dimensional semi-Riemannian manifold, ∧ k (M ) be k-forms set defined on M and dv be volume element.Hence, * : If " * " isomorphism holds the following equality for ∀α, β ∈ ∧ k (M ), α ∧ * β = α, β dv (2.9) then this transformation is called Hodge-star operator [6].
For n-dimensional semi-Riemannian manifold M , gradf denotes the gradient of f and we define as [7] In addition we define the Laplace operator on M as [7] and also we define Hessian form of differentiable function f on M as [7] H where H f (e i , e j ) = f ij .
In addition, we define the second fundamental form of vector fields U and V of nondegenerate boundary ∂M ′ = ∂M + ∪ ∂M − as follows: Definition 2.1.Let M be an n-dimensional semi-Riemannian manifold and R be a Riemannian curvature tensor of M .Let {e 1 , e 2 , ..., e n } be a semi-Riemannian orthonormal basis of T p (M ).Thus, one can write the following: where the curvature tensor field Ric is called Ricci curvature tensor field and also the value of Ric(U, V ) on pǫM is called Ricci curvature of M [7].
Taking U and V as follows in (2.13) and defining Rij as follows we have (2.15) Definition 2.2.Let M be an n-dimensional semi-Riemannian manifold and f ǫC ∞ (M, R).Hence the differential of f can be defined as follows: If {x 1 , x 2 , ..., x n } is local coordinate system on point p, then {dx 1|p , dx 2|p , ..., dx n|p } will be basis on T * p (M ).In addition there is the following relation among the components of the basis {dx 1 , dx 2 , ..., dx n } where {dx 1 , dx 2 , ..., dx n } is the dual basis of { ∂ ∂x1 , ∂ ∂x2 , ..., ∂ ∂xn } and also g ij is the inverse matrix of g ij [7].Definition 2.5.Let M be an n-dimensional semi-Riemannian manifold with nondegenerate space-like boundary and time-like unit outward normal.In addition let {u 1 , ..., u n−1 } be an orthonormal basis of T a ∂M + and D = (n 1 , ..., n n ) be a time-like unit outward normal of ∂M + .Then the equality defined in the above is called volume element of ∂M + .Definition 2.6.Let M be an n-dimensional semi-Riemannian manifold with nondegenerate time-like boundary and space-like unit outward normal.In addition let {u 1 , ..., u n−1 } be an orthonormal basis of T a ∂M − and D = (n 1 , ..., n n ) be a spacelike unit outward normal of ∂M − .Then the equality defined in the above is called volume element of ∂M − .Definition 2.7.Let U be an open set of semi-Riemannian manifold M and w 1 , w 2 , ..., w n be 1-forms on U .In addition, let w i j be connection coefficients.E. Cartan structure equations are defined as follows: 1. E. Cartan Structure Equation; where R ijkl is the component of the Riemannian-Christoffel curvature tensor [8].
Lemma 2.8.(Cartan's Lemma) Let M be an n-dimensional manifold and w i be 1-forms on M for i = 1, 2, ..., n .In addition, let λ i be the other 1-forms.Suppose that λ i and w i are linearly independent.Then Hence, for 1 ≤ i, j ≤ n, a ij = a ji and a ij ǫC ∞ (M, R), one can write the following [6] Theorem 2.9.Let M be an n-dimensional semi-Riemannian manifold and U be open subset of M .In addition, let w 1 , w 2 , ..., w n be 1-forms on U ⊂ M and f be any differentiable function.Then where i, j, k = 1, ..., n.
The differential of f is as follows: Here, by using exterior differential, we get Considering (2.18) equality here and making routine calculations, we have The Reilly's Integral Formula on Semi-Riemannian Manifolds

131
In addition, from Lemma 2.8, there are f ij functions on an open subset U of M .Hence, from f ij = f ji , the above equality can be written as follows: Using exterior differential for the (2.20) equality, we obtain n j=1 From Lemma 2.8, there are f ijk functions on an open subset U of M .Then we have On the other hand, let Here writing the similar equality of Eq. (2.22) for f ijk and considering Eq. (2.21), we get This completes the proof of theorem. 2 Definition 2.10.Let M be an n-dimensional semi-Riemannian manifold and N be a semi-Riemannian submanifold with index (v − 1).Let's consider the following isometric immersion Owing to this immersion, local semi-Riemannian orthonormal basis {e 1 , e 2 , ..., e n−1 } in the coordinate neighbourhood U of N transforms a local semi-Riemannian orthonormal basis {e 1 , e 2 , ..., e n }. (Here, e n is the time-like unit outward normal of N ).In addition, 1-forms {w 1 , w 2 , ..., w n } which are the dual basis of local semi-Riemannian orthonormal basis {e 1 , e 2 , ..., e n } transform dual 1-forms which are defined as follows of N , under this immersion, and connection coefficients w i j , 1 ≤ i, j ≤ n also transform dual connection coefficients θ i j of N which are defined as follows That is, 1-forms w i , 1 ≤ i ≤ n under the transformation of τ transform 1-forms θ 1 , θ 2 , ..., θ n−1 on N , where This 1-forms are shortly called co-frame.
On the other hand, using exterior differential for (2.26), we have dθ n = 0.In addition, let II ij = II ji that are the components of second fundamental form of N .This 1-forms are defined as follows [1]: and (2.28) Definition 2.11.Let M and M be an n-dimensional and (n-1)-dimensional semi-Riemannian manifold, respectively.Let's consider the following isometric immersion Owing to this isometric immersion, local semi-Riemannian orthonormal basis {e 1 , e 2 , ..., e n−1 } in the coordinate neighbourhood U of M transforms a local semi-Riemannian orthonormal basis {e 1 , e 2 , ..., e n } of M .(Here, e n is the space-like unit outward normal of M ).In addition, 1-forms {w 1 , w 2 , ..., w n } which are the dual basis of local semi-Riemannian orthonormal basis {e 1 , e 2 , ..., e n } transform dual 1-forms θ i which are defined as follows of M , under this immersion, and connection coefficients w i j , 1 ≤ i, j ≤ n, also transform dual connection coefficients θ i j of M which are defined as follows That is, 1-forms w i , 1 ≤ i ≤ n, under the transformation of τ transform 1-forms θ 1 , θ 2 , ..., θ n−1 on M , where

31)
The Reilly's Integral Formula on Semi-Riemannian Manifolds
On the other hand, using exterior differential for (2.31), we have dθ n = 0.In addition, let II ij = II ji that are the components of second fundamental form of M .This 1-forms are defined as follows: and such that τ is inclusion transformation as follows: Thus, we can write the following: Here, if Here, getting u instead of f n and using (2.22) and (2.25), we get In the last equality, considering (2.24) and (2.27), we have Let's consider (2.34) equality for computing f ni .According to this, we get The Reilly's Integral Formula on Semi-Riemannian Manifolds 135 Using (2.27) in the last equality, we obtain Thus, the covariant derivatives of f i , f ij , f ni are obtained for semi-Riemannian manifold with nondegenerate spacelike boundary ∂M + .Definition 2.13.Let M be an n-dimensional semi-Riemannian manifold with nondegenerate time-like boundary ∂M − .Let {e 1 , ..., e n−1 } be local semi-Riemannian orthonormal frame of ∂M − and e n be a space-like unit outward normal of ∂M − .Thus {e 1 , ..., e n } is a local semi-Riemannian orthonormal frame of M .Let {w 1 , ..., w n } be dual co-frame of this local semi-Riemannian orthonormal frame and f be a differentiable function on M .Now, getting ∂M − instead of M in definition 2.11 let's study the covariant derivative of function f .Firstly, let's compute f i , f ij , f ni on a point of ∂M − .Hence let's take such that τ is inclusion transformation as follows: Thus, we can write the following: Here, if Using exterior differential for (2.42), we get Here, getting u instead of f n and considering (2.22), (2.29) and (2.30), we obtain and using (2.32), we get Now, let's consider (2.40) equality for computing f ni .According to this, we have Using (2.32) in the last equality, we obtain The Reilly's Integral Formula on Semi-Riemannian Manifolds 137 or where β j is as (2.3).
ii) If M has ∂M − time-like boundary and D space-like unit outward normal, w ∂M− volume element of ∂M − is as follows: where ε j is as Eq.(2.4).

Reilly's Formula
Let M be an n-dimensional compact, orientable semi-Riemannian manifold with nondegenerate boundary ∂M ′ (∂M ′ = ∂M + ∪ ∂M − ).Thus, for wǫ ∧ 1 (M ) we can write the following from Stokes theorem where * is the Hodge-star operator.
Here, considering the property of Hodge-star operator, we have * w = n i,j=1

Definition 2 . 3 .Definition 2 . 4 .Remark 1 .
Let M be an n-dimensional semi-Riemannian manifold with boundary ∂M .Then the open subset ∂M + is called nondegenerate space-like boundary where unit outward normal is timelike and index of the induced nondegenerate metric is v − 1 on ∂M + .The Reilly's Integral Formula on Semi-Riemannian Manifolds 129 Let M be an n-dimensional semi-Riemannian manifold with boundary ∂M .Then the open subset ∂M − is called nondegenerate time-like boundary where unit outward normal is spacelike and index of the induced nondegenerate metric is v on ∂M − .Note that ∂M = ∂M + ∪∂M − ∪∂M 0 and ∂M + , ∂M − , ∂M 0 are pairwise disjoint subsets of ∂M .Also notice that ∂M + and ∂M − are open submanifolds of ∂M and ∂M ′ = ∂M + ∪ ∂M − can be considered as the nondegenerate boundary of M .
.33) Definition 2.12.Let M be an n-dimensional semi-Riemannian manifold with nondegenerate space-like boundary ∂M + .Let {e 1 , ..., e n−1 } be local semi-Riemannian orthonormal frame of ∂M + and e n be a time-like unit outward normal of ∂M + .Thus {e 1 , ..., e n } is a local semi-Riemannian orthonormal frame of M .Let {w 1 , ..., w n } be dual co-frame of this local semi-Riemannian orthonormal frame and f be a differentiable function on M .