Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold

Open Access
Research

Abstract

We introduce k-Ricci curvature and k-scalar curvature on lightlike hypersurfaces of a Lorentzian manifold. We establish some inequalities between the extrinsic scalar curvature and the intrinsic scalar curvature. Using these inequalities, we obtain some characterizations on lightlike hypersurfaces. We give some results with regard to curvature invariants and S ( n 1 , , n k ) Open image in new window-spaces for lightlike hypersurfaces of a Lorentzian manifold.

Keywords

Scalar Curvature Sectional Curvature Fundamental Form Plane Section Ricci Curvature 

1 Introduction

In 1873, Schläfli conjectured that every Riemann manifold could be locally considered as a submanifold of an Euclidean space with sufficient high codimension. This was proven by Janet in [1], Cartan in [2]. Friedmann extended the theorem to semi-Riemannian manifolds in [3]. Chen gave a relation between the sectional curvature and the shape operator for an n-dimensional submanifold M in a Riemannian space form R m ( c ¯ ) Open image in new window in [4] as follows:
A N > n 1 n ( c c ¯ ) I n , Open image in new window
(1.1)

where A N Open image in new window is a shape operator of M, c = inf K c ¯ Open image in new window and I n Open image in new window is an identity map. Also, Chen established a sharp inequality between the main intrinsic curvatures (the sectional curvature and the scalar curvature) and the main extrinsic curvatures (the squared mean curvature) for a submanifold in a real space form R m ( c ¯ ) Open image in new window in [5] as follows:

For each unit tangent vector X T p M n Open image in new window,
H 2 ( p ) 4 n 2 { Ric ( X ) ( n 1 ) c ¯ } , Open image in new window
(1.2)

where H 2 Open image in new window is the squared mean curvature and Ric ( X ) Open image in new window is Ricci curvature of M n Open image in new window at X.

In [6], Hong and Tripathi presented a general inequality for submanifolds of a Riemannian manifold by using (1.2). In [7], this inequality was named Chen-Ricci inequality by Tripathi.

In [8] and [9], Chen introduced a Riemannian invariant δ ( n 1 , , n k ) Open image in new window by
δ ( n 1 , , n k ) = τ ( p ) inf { τ π 1 ( p ) + + τ π k ( p ) } , Open image in new window
(1.3)

where τ ( p ) Open image in new window is scalar curvature of M, τ π j ( p ) Open image in new window is j-scalar curvature, π 1 , , π k Open image in new window run over all k mutually orthogonal subspaces of T p M Open image in new window such that dim π j = n j Open image in new window, j = 1 , , k Open image in new window. In [10], the authors gave optimal relationships among invariant δ ( n 1 , , n k ) Open image in new window, the intrinsic curvatures and the extrinsic curvatures.

Later, Chen and some authors found inequalities for submanifolds of different spaces. For example, these inequalities were studied at submanifolds of complex space forms in [11, 12, 13]. Contact versions of Chen inequalities and their applications were introduced in [7, 14, 15, 16]. In [17], Tripathi investigated these inequalities in curvature-like tensors. Furthermore, Haesen presented an optimal inequality for an m-dimensional Lorentzian manifold embedded as a hypersurface on an ( m + 1 ) Open image in new window-dimensional Ricci-flat space in [18]. The authors in [19] proved an inequality using the extrinsic and the intrinsic scalar curvature in a semi-Riemannian manifold. In [20], Chen introduced space-like submanifolds (Riemannian submanifolds) of a semi-Riemannian manifold.

As far as we know, there is no paper about Chen-like inequalities and curvature invariants in lightlike geometry. So, we introduce k-plane Ricci curvature and k-plane scalar curvature of a lightlike hypersurface of a Lorentzian manifold. Using these curvatures, we establish some inequalities and by means of these inequalities, we give some characterizations of a lightlike hypersurface on a Lorentzian manifold. Finally, we introduce the curvature invariant δ ( n 1 , , n k ) Open image in new window on lightlike hypersurfaces of a Lorentzian manifold.

2 Preliminaries

Let ( M ˜ , g ˜ ) Open image in new window be an ( n + 2 ) Open image in new window-dimensional semi-Riemannian manifold and M be a lightlike hypersurface of M ˜ Open image in new window. The radical space or the null space of T p M Open image in new window, at each point p M Open image in new window, is a one-dimensional subspace Rad T p M Open image in new window defined by
Rad T p M = { ξ T p M : g p ( ξ , X ) = 0 , X T p M } . Open image in new window
(2.1)
The complementary vector bundle S ( T M ) Open image in new window of Rad T M Open image in new window in TM is called the screen bundle of M. We note that any screen bundle is non-degenerate. Thus, we have
T M = Rad T M S ( T M ) , Open image in new window
(2.2)
where ⊥ denotes the orthogonal direct sum. The complementary vector bundle S ( T M ) Open image in new window of S ( T M ) Open image in new window is called screen transversal bundle and it has rank 2. Since Rad T M Open image in new window is a lightlike subbundle of S ( T M ) Open image in new window, there exists a unique local section N of S ( T M ) Open image in new window such that
g ˜ ( N , N ) = 0 , g ˜ ( N , ξ ) = 1 . Open image in new window
(2.3)
The Gauss and Weingarten formulas are given, respectively, by
˜ X Y = X Y + h ( X , Y ) , ˜ X N = A N X + X t N Open image in new window
(2.4)
for any X , Y Γ ( T M ) Open image in new window, where X Y , A N X Γ ( T M ) Open image in new window and h ( X , Y ) , X t N Γ ( ltr ( T M ) ) Open image in new window. If we put B ( X , Y ) = g ˜ ( h ( X , Y ) , ξ ) Open image in new window and τ ( X ) = g ˜ ( X t N , ξ ) Open image in new window, then (2.4) become
˜ X Y = X Y + B ( X , Y ) N , ˜ X N = A N X + τ ( X ) N , Open image in new window
(2.5)

where B and A N Open image in new window are called the second fundamental form and the shape operator of the lightlike hypersurface M. The induced connection ∇ on M is not metric connection but ∇ is torsion free [21].

If B = 0 Open image in new window, then the lightlike hypersurface M is called totally geodesic in M ˜ Open image in new window. A point p M Open image in new window is said to be umbilical if
B ( X , Y ) p = H g p ( X , Y ) , X , Y T p M , Open image in new window

where H R Open image in new window. M is called totally umbilical in M ˜ Open image in new window if every point of M is umbilical [21].

The mean curvature μ of M with respect to an { e 1 , , e n } Open image in new window orthonormal basis of Γ ( S ( T M ) ) Open image in new window is defined in [22] as follows:
μ = tr ( B ) n = 1 n i = 1 n ε i B ( e i , e i ) , g ( e i , e i ) = ε i . Open image in new window
(2.6)
Let P be a projection of S ( T M ) Open image in new window on M. From (2.2), we have
X P Y = X P Y + h ( X , P Y ) = X Y + C ( X , P Y ) ξ , X , Y Γ ( T M ) , Open image in new window
(2.7)
X ξ = A ξ X τ ( X ) ξ , Open image in new window
(2.8)

where X P Y Open image in new window and A ξ X Open image in new window belong to Γ ( S ( T M ) ) Open image in new window. Here Open image in new window, C and A ξ Open image in new window are called the induced connection, the local second fundamental form and the local shape operator on S ( T M ) Open image in new window, respectively.

From (2.5) and (2.7) one has
B ( X , Y ) = g ( A ξ X , Y ) , Open image in new window
(2.9)
C ( X , P Y ) = g ( A N X , P Y ) . Open image in new window
(2.10)
Using (2.9) we have
B ( X , ξ ) = 0 , X Γ ( T M | U ) . Open image in new window
A lightlike hypersurface ( M , g ) Open image in new window of a semi-Riemannian manifold ( M ˜ , g ˜ ) Open image in new window is called screen locally conformal if the shape operators A N Open image in new window and A ξ Open image in new window of M and S ( T M ) Open image in new window, respectively, are related by
A N = φ A ξ , Open image in new window
(2.11)

where φ is a non-vanishing smooth function on a neighborhood U on M [23]. In particular, M is called screen homothetic if φ is a non-zero constant.

We denote the Riemann curvature tensors of M ˜ Open image in new window and M by R ˜ Open image in new window and R, respectively. The Gauss-Codazzi type equations for M are given as follows:
g ˜ ( R ˜ ( X , Y ) Z , P U ) = g ( R ( X , Y ) Z , P U ) + B ( X , Z ) C ( Y , P U ) B ( Y , Z ) C ( X , P U ) , Open image in new window
(2.12)
g ˜ ( R ˜ ( X , Y ) Z , ξ ) = ( X B ) ( Y , Z ) ( Y B ) ( X , Z ) + B ( Y , Z ) τ ( X ) B ( X , Z ) τ ( Y ) , Open image in new window
(2.13)
g ˜ ( R ˜ ( X , Y ) Z , N ) = g ( R ( X , Y ) Z , N ) , Open image in new window
(2.14)
g ˜ ( R ˜ ( X , Y ) P Z , N ) = ( X C ) ( Y , P Z ) ( Y C ) ( X , P Z ) + τ ( Y ) C ( X , P Z ) τ ( X ) C ( Y , P Z ) Open image in new window
(2.15)

for any X , Y , Z , U Γ ( T M ) Open image in new window [21].

Let p M Open image in new window and Π = sp { e i , e j } Open image in new window be a two-dimensional non-degenerate plane of T p M Open image in new window. The number
K i j = g ( R ( e j , e i ) e i , e j ) g ( e i , e i ) g ( e j , e j ) g ( e i , e j ) 2 Open image in new window

is called the sectional curvature at p M Open image in new window. Since the screen second fundamental form C is not symmetric, the sectional curvature K i j Open image in new window of a lightlike submanifold is not symmetric, that is, K i j K j i Open image in new window.

Let p M Open image in new window and ξ be a null vector of T p M Open image in new window. A plane Π of T p M Open image in new window is called a null plane if it contains ξ and e i Open image in new window such that g ˜ ( ξ , e i ) = 0 Open image in new window and g ˜ ( e i , e i ) 0 Open image in new window. The null sectional curvature of Π is given in [24] as follows:
K i null = g ( R p ( e i , ξ ) ξ , e i ) g p ( e i , e i ) . Open image in new window
The Ricci tensor Ric ˜ Open image in new window of M ˜ Open image in new window and the induced Ricci type tensor R ( 0 , 2 ) Open image in new window of M are defined by
Ric ˜ ( X , Y ) = trace { Z R ˜ ( Z , X ) Y } , X , Y Γ ( T M ˜ ) , R ( 0 , 2 ) ( X , Y ) = trace { Z R ( Z , X ) Y } , X , Y Γ ( T M ) . Open image in new window
(2.16)
Let { e 1 , , e n } Open image in new window be an orthonormal frame of Γ ( S ( T M ) ) Open image in new window. In this case,
R ( 0 , 2 ) ( X , Y ) = j = 1 n ε j g ( R ( e j , X ) Y , e j ) + g ˜ ( R ( ξ , X ) Y , N ) , Open image in new window
(2.17)
where ε j Open image in new window denotes the causal character (∓1) of a vector field e j Open image in new window. Ricci curvature of a lightlike hypersurface is not symmetric. Thus, Einstein hypersurfaces are not defined on any lightlike hypersurface. If M admits that an induced symmetric Ricci tensor Ric and Ricci tensor satisfy
Ric ( X , Y ) = k g ( X , Y ) , X , Y Γ ( T M ) , Open image in new window
(2.18)

where k is a constant, then M is called an Einstein hypersurface [23].

Let M be a lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window, replacing X by ξ and using (2.12), (2.13) and (2.14)
R ( 0 , 2 ) ( ξ , ξ ) = j = 1 n g ( R ( e j , ξ ) ξ , e j ) g ˜ ( R ( ξ , ξ ) ξ , N ) = j = 1 n K j null . Open image in new window
(2.19)
Thus, we have
i = 1 n R ( 0 , 2 ) ( e i , e i ) = i = 1 n { j = 1 n g ( R ( e j , e i ) e i , e j ) } + i = 1 n g ˜ ( R ( ξ , e i ) e i , N ) . Open image in new window
(2.20)
Adding (2.19) and (2.20), we obtain a scalar τ given as follows [25]:
τ = R ( 0 , 2 ) ( ξ , ξ ) + i = 1 n R ( 0 , 2 ) ( e i , e i ) = i , j = 1 n K i j + i = 1 n K i null + K i N , Open image in new window
(2.21)

where K i N = g ˜ ( R ( ξ , e i ) e i , N ) Open image in new window for i { 1 , , n } Open image in new window.

3 k-Ricci curvature and k-scalar curvature

Let M be an ( n + 1 ) Open image in new window-dimensional lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window and let { e 1 , , e n , ξ } Open image in new window be a basis of Γ ( T M ) Open image in new window where { e 1 , , e n } Open image in new window is an orthonormal basis of Γ ( S ( T M ) ) Open image in new window. For k n Open image in new window, we set π k , ξ = sp { e 1 , , e k , ξ } Open image in new window is a ( k + 1 ) Open image in new window-dimensional degenerate plane section and π k = sp { e 1 , , e k } Open image in new window is a k-dimensional non-degenerate plane section.

We say that k-degenerate Ricci curvature and k-Ricci curvature at unit vector X Γ ( T M ) Open image in new window are as follows:
Ric π k , ξ ( X ) = R ( 0 , 2 ) ( X , X ) = j = 1 k g ( R ( e j , X ) X , e j ) + g ˜ ( R ( ξ , X ) X , N ) , Open image in new window
(3.1)
Ric π k ( X ) = R ( 0 , 2 ) ( X , X ) = j = 1 k g ( R ( e j , X ) X , e j ) , Open image in new window
(3.2)
respectively. Furthermore, we say that k-degenerate scalar curvature and k-scalar curvature at p M Open image in new window are as follows:
τ π k , ξ ( p ) = i , j = 1 k K i j + i = 1 k K i null + K i N , Open image in new window
(3.3)
τ π k ( p ) = i , j = 1 k K i j , Open image in new window
(3.4)
respectively. For k = 2 Open image in new window, Π 1 , ξ = sp { e 1 , ξ } Open image in new window, then we have
Ric Π 1 , ξ ( e 1 ) = K 1 N , Open image in new window
and
τ Π 2 ( p ) = K 1 null + K 1 N . Open image in new window
For k = n Open image in new window, π n = sp { e 1 , , e n } = Γ ( S ( T M ) ) Open image in new window, then
Ric S ( T M ) ( e 1 ) = Ric π n ( e 1 ) = j = 1 n K 1 j = K 12 + + K 1 n , Open image in new window
(3.5)
and
τ S ( T M ) ( p ) = i , j = 1 n K i j . Open image in new window
(3.6)
We say that screen Ricci curvature and screen scalar curvature are Ric S ( T M ) ( e 1 ) Open image in new window and τ S ( T M ) ( p ) Open image in new window, respectively. From (2.12) we can write
τ S ( T M ) ( p ) = τ ˜ S ( T M ) ( p ) + i , j = 1 n B i i C j j B i j C j i , Open image in new window
(3.7)

where B i j = B ( e i , e j ) Open image in new window and C i j = C ( e i , e j ) Open image in new window for i , j { 1 , , n } Open image in new window.

Also, the components of the second fundamental form B and the screen second fundamental form C satisfy
i , j = 1 n B i j C j i = 1 2 { i , j = 1 n ( B i j + C j i ) 2 i , j = 1 n ( B i j ) 2 + ( C j i ) 2 } , Open image in new window
(3.8)
and
i , j B i i C j j = 1 2 { ( i , j B i i + C j j ) 2 ( i B i i ) 2 ( j C j j ) 2 } . Open image in new window
(3.9)
Theorem 3.1 Let M be an ( n + 1 ) Open image in new window-dimensional lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window. Then:
  1. (a)
    τ S ( T M ) ( p ) τ ˜ S ( T M ) ( p ) + n μ ( trace A N ) + 1 4 i , j ( B i j C j i ) 2 . Open image in new window
    (3.10)
     
The equality holds for all p M Open image in new window if and only if either M is a screen homothetic lightlike hypersurface with φ = 1 Open image in new window or M is a totally geodesic lightlike hypersurface.
  1. (b)
    τ S ( T M ) ( p ) τ ˜ S ( T M ) ( p ) + n μ ( trace A N ) 1 2 i , j ( B i j + C j i ) 2 . Open image in new window
    (3.11)
     
The equality holds for all p M Open image in new window if and only if either M is a screen homothetic lightlike hypersurface with φ = 1 Open image in new window or M is a totally geodesic lightlike hypersurface.
  1. (c)

    The equalities case of (3.10) and (3.11) hold at p M Open image in new window if and only if p is a totally geodesic point.

     
Proof Using (3.7) and (3.8), we get
τ S ( T M ) ( p ) = τ ˜ S ( T M ) ( p ) + i , j = 1 n B i i C j j 1 2 i , j = 1 n ( B i j + C j i ) 2 + 1 2 i , j = 1 n ( B i j ) 2 + ( C j i ) 2 . Open image in new window
(3.12)
Since
1 2 ( B i j 2 + C j i 2 ) = 1 4 ( B i j + C j i ) 2 + 1 4 ( B i j C j i ) 2 , Open image in new window
then
1 2 { i , j = 1 n ( B i j + C j i ) 2 + i , j = 1 n ( B i j ) 2 + ( C j i ) 2 } = 1 2 i , j ( B i j + C j i ) 2 + 1 4 i , j ( B i j C j i ) 2 . Open image in new window
(3.13)
If we put (3.13) in (3.12), we obtain
τ S ( T M ) ( p ) = τ ˜ S ( T M ) ( p ) + i , j = 1 n B i i C j j 1 2 i , j ( B i j + C j i ) 2 + 1 4 i , j ( B i j C j i ) 2 , Open image in new window
(3.14)

which yields (3.10) and (3.11). From (3.10), (3.11) and (3.14) it is easy to get (a), (b) and (c) statements. □

Corollary 3.2 Let M be an ( n + 1 ) Open image in new window-dimensional lightlike hypersurface of a Lorentzian space form M ˜ ( c ) Open image in new window. Then:
  1. (a)
    τ S ( T M ) ( p ) n ( n 1 ) c + n μ ( trace A N ) + 1 4 i , j ( B i j C j i ) 2 . Open image in new window
    (3.15)
     
  2. (b)
    τ S ( T M ) ( p ) n ( n 1 ) c + n μ ( trace A N ) 1 2 i , j ( B i j + C j i ) 2 . Open image in new window
    (3.16)
     
Corollary 3.3 Let M be an ( n + 1 ) Open image in new window-dimensional screen homothetic lightlike hypersurface of a Lorentzian space form M ˜ ( c ) Open image in new window. Then:
  1. (a)
    τ S ( T M ) ( p ) n ( n 1 ) c + φ n 2 μ 2 + ( φ 1 ) 4 i , j ( B i j ) 2 . Open image in new window
    (3.17)
     
  2. (b)
    τ S ( T M ) ( p ) n ( n 1 ) c + φ n 2 μ 2 ( φ 1 ) 2 i , j ( B i j ) 2 . Open image in new window
    (3.18)
     
Theorem 3.4 Let M be an ( n + 1 ) Open image in new window-dimensional lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window. Then
τ S ( T M ) ( p ) τ ˜ S ( T M ) ( p ) + 1 2 ( trace A ¯ ) 2 1 2 ( trace A N ) 2 1 2 i , j ( B i j + C j i ) 2 + 1 4 i , j ( B i j C j i ) 2 , Open image in new window
(3.19)
where
A ¯ = ( B 11 + C 11 B 12 + C 21 B 1 n + C n 1 B 21 + C 12 B 22 + C 22 B 2 n + C n 2 B n 1 + C 1 n B n 2 + C 2 n B n n + C n n ) . Open image in new window
(3.20)

The equality of (3.19) holds for all p M Open image in new window if and only if M is minimal.

Proof From (3.14) and (3.9) we get
τ S ( T M ) ( p ) = τ ˜ S ( T M ) ( p ) + 1 2 { ( i , j B i i + C j j ) 2 ( i B i i ) 2 ( j C j j ) 2 } 1 2 i , j ( B i j + C j i ) 2 + 1 4 i , j ( B i j C j i ) 2 , Open image in new window
(3.21)

which implies (3.19).

The equality of (3.19) satisfies then
i B i i = 0 . Open image in new window
(3.22)

This shows that M is minimal. □

By Theorem 3.4 we get the following corollaries.

Corollary 3.5 Let M be an ( n + 1 ) Open image in new window-dimensional lightlike hypersurface of a Lorentzian space form M ˜ ( c ) Open image in new window. Then
τ S ( T M ) ( p ) n ( n 1 ) c + 1 2 ( trace A ¯ ) 2 1 2 ( trace A N ) 2 1 2 i , j ( B i j + C j i ) 2 + 1 4 i , j ( B i j C j i ) 2 , Open image in new window
(3.23)

where A ¯ Open image in new window is equal to (3.20). The equality of (3.23) holds for all p M Open image in new window if and only if M is minimal.

Corollary 3.6 Let M be an ( n + 1 ) Open image in new window-dimensional screen homothetic lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window. Then
τ S ( T M ) ( p ) τ ˜ S ( T M ) ( p ) + ( 2 φ + 1 ) 2 n 2 μ 2 ( φ 2 + 6 φ + 1 ) 4 i , j ( B i j ) 2 . Open image in new window
(3.24)

The equality of (3.24) holds for all p M Open image in new window if and only if M is minimal.

Now, we shall need the following lemma.

Lemma 3.7 [26]

If a 1 , , a n Open image in new window are n-real numbers ( n > 1 Open image in new window), then
1 n ( i = 1 n a i ) 2 i = 1 n a i 2 , Open image in new window
(3.25)

with equality if and only if a 1 = = a n Open image in new window.

Theorem 3.8 Let M be an ( n + 1 ) Open image in new window-dimensional ( n > 1 Open image in new window) lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window. Then
τ S ( T M ) ( p ) τ ˜ S ( T M ) ( p ) + n 1 2 n ( trace A ¯ ) 2 1 2 { ( trace A N ) 2 + n 2 μ 2 } + 1 4 i , j ( B i j C j i ) 2 1 2 i j ( B i j + C j i ) 2 , Open image in new window
(3.26)

where A ¯ Open image in new window is equal to (3.20).

The equality case of (3.26) holds for all p M Open image in new window if and only if n μ = trace A N Open image in new window.

Proof From (3.21)
τ S ( T M ) ( p ) = τ ˜ S ( T M ) ( p ) + 1 2 { ( trace A ¯ ) 2 ( trace A N ) 2 n 2 μ 2 } 1 2 i ( B i i + C i i ) 2 1 2 i j ( B i j + C j i ) 2 + 1 4 i , j ( B i j C j i ) 2 . Open image in new window
(3.27)
Using Lemma 3.7 and equality (3.27), we have
τ S ( T M ) ( p ) τ ˜ S ( T M ) ( p ) + 1 2 { ( trace A ¯ ) 2 ( trace A N ) 2 n 2 μ 2 } 1 2 n ( i B i i + C i i ) 2 1 2 i j ( B i j + C j i ) 2 + 1 4 i , j ( B i j C j i ) 2 , Open image in new window
(3.28)

which implies (3.26).

The equality case of (3.26) satisfies then
B 11 + C 11 = = B n n + C n n . Open image in new window
(3.29)
From (3.29) we get
( 1 n ) B 11 + B 22 + + B n n + ( 1 n ) C 11 + C 22 + + C n n = 0 , B 11 + ( 1 n ) B 22 + + B n n + C 11 + ( 1 n ) C 22 + + C n n = 0 , B 11 + B 22 + + ( 1 n ) B n n + C 11 + C 22 + + ( 1 n ) C n n = 0 . Open image in new window
By the above equations, we obtain
( n 1 ) 2 ( trace A N + n μ ) = 0 . Open image in new window
(3.30)

Since n 1 Open image in new window, n μ = trace A N Open image in new window. □

From Theorem 3.8 we get the following corollaries.

Corollary 3.9 Let M be an ( n + 1 ) Open image in new window-dimensional ( n > 1 Open image in new window) lightlike hypersurface of a Lorentzian space form M ˜ ( c ) Open image in new window. Then
τ S ( T M ) ( p ) n ( n 1 ) c + n 1 2 n ( trace A ¯ ) 2 1 2 { n 2 μ 2 + ( trace A N ) 2 } + 1 4 i , j ( B i j C j i ) 2 1 2 i j ( B i j + C j i ) 2 , Open image in new window
(3.31)

where A ¯ Open image in new window is equal to (3.20).

The equality case of (3.31) holds for all p M Open image in new window if and only if n μ = trace A N Open image in new window.

Corollary 3.10 Let M be an ( n + 1 ) Open image in new window-dimensional ( n > 1 Open image in new window) screen homothetic lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window. Then
τ S ( T M ) ( p ) τ ˜ S ( T M ) ( p ) + φ n 2 μ 2 ( φ + 1 ) 2 2 n μ 2 + ( φ 1 ) 2 4 i ( B i i ) 2 ( φ 2 + 6 φ + 1 ) 4 i j ( B i j ) 2 . Open image in new window
(3.32)

The equality case of (3.32) holds for all p M Open image in new window if and only if either φ = 1 Open image in new window or M is minimal.

4 Curvature invariants on lightlike hypersurfaces

Definition 4.1 For an integer k 0 Open image in new window, let S ( n , k ) Open image in new window be the finite set which consists of k-tuples ( n 1 , , n k ) Open image in new window of integers ≥2 satisfying n 1 < n Open image in new window and n 1 + + n k n Open image in new window. Denote by S ( n ) Open image in new window the set of all unordered k-tuples with k 0 Open image in new window for a fixed positive integer n.

For each k-tuple ( n 1 , , n k ) S ( n ) Open image in new window, the two sequences of curvature invariants S ( n 1 , , n k ) ( p ) Open image in new window and S ˆ ( n 1 , , n k ) ( p ) Open image in new window are defined by, respectively,
S ( n 1 , , n k ) ( p ) = inf { τ ( Π n 1 ) + + τ ( Π n k ) } , S ˆ ( n 1 , , n k ) ( p ) = sup { τ ( Π n 1 ) + + τ ( Π n k ) } , Open image in new window
where Π n 1 , , Π n k Open image in new window are k-dimensional mutually orthogonal subspaces of T p M Open image in new window such that dim Π n j = n j Open image in new window, j = 1 , , k Open image in new window.
δ ( n 1 , , n k ) ( p ) = τ ( p ) S ( n 1 , , n k ) ( p ) , δ ˆ ( n 1 , , n k ) ( p ) = τ ( p ) S ˆ ( n 1 , , n k ) ( p ) . Open image in new window

We call a lightlike hypersurface an S ( n 1 , , n k ) Open image in new window space if it satisfies S ( n 1 , , n k ) = S ˆ ( n 1 , , n k ) Open image in new window.

Theorem 4.2 Let M be a lightlike hypersurface of an ( n + 2 ) Open image in new window-dimensional Lorentzian manifold M ˜ Open image in new window. Then M is an S ( n ) Open image in new window space if and only if the scalar curvature of M is constant.

Proof Let { e 1 , , e n } Open image in new window be an orthonormal frame of Γ ( S ( T M ) ) Open image in new window. Let us choose n-dimensional plane sections such that
π n 1 , ξ 1 = sp { e 2 , , e n , ξ } T p M , π n 1 , ξ n = sp { e 1 , , e n 1 , ξ } T p M , π n = sp { e 1 , , e n } T p M . Open image in new window
Thus, from (3.3) and (3.4), we obtain
τ π n 1 , ξ 1 ( p ) = i , j = 2 n K i j + K i null + K i N , τ π n 1 , ξ n ( p ) = i , j = 1 n 1 K i j + K i null + K i N , τ π n ( p ) = i , j = 1 n K i j = τ S ( T M ) ( p ) . Open image in new window
If M is an S ( n ) Open image in new window space, then we can write
τ π n 1 , ξ 1 ( p ) = τ ( p ) Ric ( e 1 ) K 1 null = c , τ π n 1 , ξ n ( p ) = τ ( p ) Ric ( e n ) K n null = c , τ π n ( p ) = τ ( p ) i = 1 n K i null + K i N = c . Open image in new window
Therefore, we have
Ric ( e 1 ) + K 1 null = = Ric ( e n ) + K n null = i = 1 n K i null + K i N . Open image in new window
(4.1)
From (4.1) we get
Ric S ( T M ) ( e 1 ) + + Ric S ( T M ) ( e n ) = ( n 1 ) i = 1 n K i null + K i N . Open image in new window
(4.2)
Using (4.2) we have
τ S ( T M ) ( p ) = ( n 1 ) ( τ ( p ) τ S ( T M ) ( p ) ) . Open image in new window
Thus, we obtain
τ ( p ) = ( 2 n 1 n ) c , Open image in new window
(4.3)

which shows that τ ( p ) Open image in new window is constant, which completes the proof. □

Remark 4.3 We note that if an n-dimensional non-degenerate manifold is an S ( n ) Open image in new window space, then it is an Einstein space (see [10]). On the other hand, if a degenerate hypersurface of a lightlike hypersurface is an S ( n ) Open image in new window space, then it has constant scalar curvature. Thus, the curvature invariants on degenerate submanifolds give different characterizations from the curvature invariants on non-degenerate submanifolds.

Keeping in view (4.2), we get the following corollary immediately.

Corollary 4.4 Let M ( c ) Open image in new window be an n-dimensional lightlike hypersurface with constant sectional curvature c. M ( c ) Open image in new window is an S ( n ) Open image in new window space if and only if i = 1 n K i N = 0 Open image in new window.

Now, we prove the following.

Theorem 4.5 Let M be a lightlike hypersurface of an ( n + 2 ) Open image in new window-dimensional Lorentzian manifold M ˜ Open image in new window. If M is an S ( j ) Open image in new window space for 2 j < n Open image in new window, then M is also S ( j + 1 ) Open image in new window space.

Proof For the proof of the theorem, we use the induction method. Firstly, we show that the claim of the theorem is true for j = 2 Open image in new window. Suppose that M is an S ( 2 ) Open image in new window space. Let us choose any two-dimensional plane sections of T p M Open image in new window as Π 1 , ξ 1 = sp { e 1 , ξ } Open image in new window, Π 2 = { e 1 , e 2 } Open image in new window, Π 1 , ξ 2 = sp { e 2 , ξ } Open image in new window. In that case,
τ Π 1 , ξ 1 ( p ) = K 1 null + K 1 N = c , τ Π 2 ( p ) = K 12 + K 21 = c , τ Π 1 , ξ 2 ( p ) = K 2 null + K 2 N = c . Open image in new window
Now, let us choose three-dimensional plane sections of T p M Open image in new window as π 3 = sp { e 1 , e 2 , e 3 } Open image in new window, π 2 , ξ 1 = sp { e 1 , e 2 , ξ } Open image in new window. If we show that τ π 2 , ξ 1 ( p ) = τ π 3 ( p ) = constant Open image in new window, then M is an S ( 3 ) Open image in new window-space
τ π 2 , ξ 1 ( p ) = K 12 + K 21 + i = 1 2 K i null + K i N = 3 c , Open image in new window
and
τ π 3 ( p ) = K 12 + K 21 + K 13 + K 31 + K 23 + K 32 = 3 c . Open image in new window

Therefore, M is an S ( 3 ) Open image in new window space.

Now, we show that the claim of the theorem is true for n = k Open image in new window.

Let us choose any k-dimensional plane sections of T p M Open image in new window as π k 1 , ξ 1 = sp { e 2 , e 3 , , e k , ξ } Open image in new window, π k 1 , ξ 2 = sp { e 1 , e 3 , , e k , ξ } Open image in new window, …, π k 1 , ξ k = sp { e 1 , e 2 , , e k 1 , ξ } Open image in new window, π k = sp { e 1 , e 2 , , e k } Open image in new window. Then
τ π k 1 , ξ 1 ( p ) = i , j = 2 k K i j + i = 2 k K i null + K i N , τ π k 1 , ξ k ( p ) = i , j = 1 k 1 K i j + i = 1 k 1 K i null + K i N , τ π k ( p ) = i , j = 1 k K i j . Open image in new window
From the above equations, we have
i = 1 k K i null + K i N = 2 c k 1 . Open image in new window
Let us choose ( k + 1 ) Open image in new window-dimensional plane sections of T p M Open image in new window as π k , ξ = sp { e 1 , , e k , ξ } Open image in new window, π k + 1 = sp { e 1 , , e k , e k + 1 } Open image in new window, then
τ π k , ξ ( p ) = i , j = 1 K i j + i = 1 k K i null + K i N = c + 2 c k 1 = ( k + 1 k 1 ) c . Open image in new window
(4.4)
Using in a similar way a special case j = 2 Open image in new window, we obtain
τ π k + 1 ( p ) = ( k + 1 k 1 ) c . Open image in new window
(4.5)

From (4.4) and (4.5) M is an S ( k + 1 ) Open image in new window space. □

Theorem 4.6 Let M be a lightlike hypersurface of an ( n + 2 ) Open image in new window-dimensional Lorentzian manifold M ˜ Open image in new window. Let { e 1 , , e n , ξ } Open image in new window be an orthonormal basis of p M Open image in new window. If M is an S ( n 1 ) Open image in new window space, then Ric ( e 1 ) = = Ric ( e n ) = constant Open image in new window and K 1 null = = K n null Open image in new window.

Proof Let M be an S ( n 1 ) Open image in new window space and π n 2 , ξ 1 = sp { e 2 , , e n 1 , ξ } Open image in new window, π n 2 , ξ 2 = sp { e 1 , e 3 , , e n 1 , ξ } Open image in new window,…, π n 2 , ξ n 1 = sp { e 1 , e 2 , , e n 2 , ξ } Open image in new window, π n 1 = sp { e 1 , e 2 , , e n 1 } Open image in new window be ( n 1 ) Open image in new window-dimensional plane sections of T p M Open image in new window. Then
τ π n 2 , ξ 1 ( p ) = τ ( p ) Ric ( e 1 ) Ric ( e n ) K 1 null K n null = c , τ π n 2 , ξ 2 ( p ) = τ ( p ) Ric ( e 2 ) Ric ( e n ) K 1 null K n null = c , τ π n 2 , ξ n 1 ( p ) = τ ( p ) Ric ( e n 1 ) Ric ( e n ) K n 1 null K n null = c , τ π n 1 ( p ) = τ ( p ) Ric ( e n ) i = 1 n K i null + K i N = c . Open image in new window
If we sum the above equations side to side and take into consideration Theorem 4.5, we have
Ric ( e 1 ) + + Ric ( e n ) + ( n 1 ) Ric ( e n ) + i = 1 n K i null + i = 1 n 1 K i null + K i N = constant . Open image in new window
Therefore, we obtain
Ric S ( T M ) ( e 1 ) + + Ric S ( T M ) ( e n ) + ( n 1 ) Ric ( e n ) + i = 1 n K i null + K i N + i = 1 n 1 K i null + K i N = constant . Open image in new window
Taking into account upper equations, we get
τ S ( T M ) ( p ) + ( n 1 ) Ric ( e n ) + τ ( p ) τ S ( T M ) ( p ) + τ π n 1 , ξ ( p ) τ π n 1 ( p ) = constant , Open image in new window

where π n 1 , ξ = sp { e 1 , , e n 1 , ξ } Open image in new window and π n 1 = sp { e 1 , , e n 1 } Open image in new window. Using Theorem 4.2 and Theorem 4.5, we obtain Ric ( e n ) = constant Open image in new window. In addition to this, from (4.1), Theorem 4.2 and Theorem 4.5, we have K 1 null = = K n null Open image in new window, which completes the proof of the theorem. □

In [25], Duggal restricted a lightlike hypersurface M (labeled by M 0 Open image in new window) of genus zero with screen distribution S ( T M ) 0 Open image in new window. He denoted this type of a lightlike hypersurface by C [ M 0 ] = [ ( M 0 , g 0 , S ( T M ) 0 ) ] Open image in new window a class of lightlike hypersurfaces of genus zero such that
  1. (a)

    M 0 Open image in new window admits a canonical screen distribution S ( T M ) 0 Open image in new window that induces a canonical lightlike transversal vector bundle N 0 Open image in new window,

     
  2. (b)

    M 0 Open image in new window admits an induced symmetric Ricci tensor, denoted by Ric0.

     

From above information, we get the following theorem immediately.

Theorem 4.7 Let M 0 Open image in new window, a member of C [ M 0 ] Open image in new window, be an ( 2 n + 1 ) Open image in new window-dimensional lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window. If M 0 Open image in new window is an Einstein hypersurface, then
τ π n , ξ ( p ) τ π n , ξ ( p ) = i = 1 n K i null i = n + 1 2 n K i null , Open image in new window
(4.6)

where π n , ξ Open image in new window is any ( n + 1 ) Open image in new window-dimensional null section of T p M 0 Open image in new window and π n , ξ Open image in new window denotes the orthogonal complement π n , ξ Open image in new window in T p M 0 Open image in new window.

Proof Let us choose an orthonormal basis { e 1 , , e 2 n } Open image in new window at p such that π n , ξ Open image in new window is spanned by { e 1 , , e n , ξ } Open image in new window. If M 0 Open image in new window is an Einstein hypersurface, then the Ricci curvature of M 0 Open image in new window satisfies
Ric ( e 1 ) + + Ric ( e n ) = Ric ( e n + 1 ) + + Ric ( e 2 n ) . Open image in new window
From (2.17) we have
i = 1 n j = 1 2 n K i j + i = 1 n K i N = i = n + 1 2 n j = 1 2 n K i j + i = n + 1 2 n K i N , Open image in new window
so, we get
i , j = 1 n K i j + i = 1 n K i N = i , j = n + 1 n K i j + i = n + 1 n K i N , Open image in new window

which is equivalent to (4.6). □

Now, we introduce these invariants as some special cases, and we get interesting characterizations on lightlike hypersurfaces as follows.

Theorem 4.8 Let M be an S ( n 1 , n 1 ) Open image in new window-space. Then:
  1. (a)

    If n 1 = 2 Open image in new window, then M is an S ( 3 ) Open image in new window-space.

     
  2. (b)
    If n 1 2 Open image in new window, then M is not necessary an S ( n 1 + 1 ) Open image in new window-space. If
    i = 1 n 1 K i null + K i N = constant , Open image in new window
     

then M is an S ( n 1 + 1 ) Open image in new window-space.

Proof (a) n 1 = 2 Open image in new window, let us choose any two-dimensional plane sections of T p M Open image in new window as Π 1 , ξ 1 = sp { e 1 , ξ } Open image in new window, Π 1 , ξ 2 = sp { e 2 , ξ } Open image in new window, Π 2 = sp { e 1 , e 2 } Open image in new window. Then
τ Π 1 , ξ 1 ( p ) = K 1 null + K 1 N , τ Π 1 , ξ 2 ( p ) = K 2 null + K 2 N , τ Π 2 ( p ) = K 12 + K 21 . Open image in new window
If M is an S ( 2 , 2 ) Open image in new window space, then
τ Π 1 , ξ 1 ( p ) + τ Π 1 , ξ 2 ( p ) = i = 1 2 K i null + K i N = c , τ Π 1 , ξ 1 ( p ) + τ Π 2 ( p ) = K 12 + K 21 + K 1 null + K 1 N = c , τ Π 1 , ξ 2 ( p ) + τ Π 2 ( p ) = K 12 + K 21 + K 2 null + K 2 N = c . Open image in new window
From the above equations, we have
τ π 2 , ξ ( p ) = 3 c 2 , Open image in new window
(4.7)

where π 2 , ξ = sp { e 1 , e 2 , ξ } Open image in new window is a three-dimensional null plane section of T p M Open image in new window.

Now, let us choose any two-dimensional plane sections of T p M Open image in new window as Π 2 1 = sp { e 1 , e 2 } Open image in new window, Π 2 2 = sp { e 1 , e 3 } Open image in new window, Π 2 3 = sp { e 2 , e 3 } Open image in new window. Since M is an S ( 2 , 2 ) Open image in new window-space, we can write
K 12 + K 21 + K 13 + K 31 + K 23 + K 32 + K 12 + K 21 + K 13 + K 31 + K 23 + K 32 = 2 τ ( π 3 ) = 3 c . Open image in new window
Therefore,
τ ( π 4 ) = 3 c 2 , Open image in new window
(4.8)
where π 3 = sp { e 1 , e 2 , e 3 } Open image in new window is a three-dimensional non-degenerate plane section of T p M Open image in new window. From (4.7) and (4.8) we see that M is an S ( 3 ) Open image in new window-space.
  1. (b)
    We show that the claim of the theorem is true for n 1 = 3 Open image in new window. Let us choose any three-dimensional plane section of T p M Open image in new window as π 2 , ξ 1 = sp { e 1 , e 2 , ξ } Open image in new window, π 2 , ξ 2 = sp { e 2 , e 3 , ξ } Open image in new window, π 2 , ξ 3 = sp { e 1 , e 3 , ξ } Open image in new window. If M is an S ( 3 , 3 ) Open image in new window-space, then
    3 c = 2 ( τ π 2 , ξ 1 ( p ) + τ π 2 , ξ 2 ( p ) + τ π 2 , ξ 3 ( p ) ) = 2 τ π 3 , ξ ( p ) + 2 i = 1 3 K i null + K i N , Open image in new window
    (4.9)
     

where π 3 , ξ = sp { e 1 , e 2 , e 3 , ξ } T p M Open image in new window. Consider (4.9), we obtain the proof of (b) condition is true. □

The proof of a general case has been seen using the same way as the special case n 1 = 3 Open image in new window.

Theorem 4.9 Let M be a ( 2 n + 1 ) Open image in new window-dimensional lightlike hypersurface of a Lorentzian manifold M ˜ Open image in new window.
  1. (a)

    If inf { τ ( π n ) + τ ( π n + 1 ) } > 0 Open image in new window, then τ ( p ) > 0 Open image in new window.

     
  2. (b)

    If sup { τ ( π n ) + τ ( π n + 1 ) } < 0 Open image in new window, then τ ( p ) < 0 Open image in new window.

     
Proof Let T p M = sp { e 1 , , e 2 n , ξ } Open image in new window. We suppose that inf { τ ( π n ) + τ ( π n + 1 ) } > 0 Open image in new window. By straightforward computation, we have
C ( 2 n 2 , n 2 ) + C ( 2 n 2 , n ) τ S ( T M ) ( p ) + C ( 2 n 1 , n 1 ) i = 1 2 n K i null + K i N > 0 , Open image in new window
(4.10)
and
C ( 2 n 2 , n 3 ) + C ( 2 n 2 , n 1 ) τ S ( T M ) ( p ) + C ( 2 n , n ) i = 1 2 n K i null + K i N > 0 . Open image in new window
(4.11)
Summing up (4.10) and (4.11), we get
C ( 2 n , n ) τ S ( T M ) ( p ) + C ( 2 n , n ) i = 1 2 n K i null + K i N > 0 , Open image in new window
(4.12)

which shows that C ( 2 n , n ) τ ( p ) > 0 Open image in new window. Therefore, τ ( p ) > 0 Open image in new window which is a proof of the statement (a).

Now, we suppose that sup { τ ( π n ) + τ ( π n + 1 ) } > 0 Open image in new window. Following a similar way in the proof of statement (a), we have
C ( 2 n , n ) τ S ( T M ) ( p ) + C ( 2 n , n ) i = 1 2 n K i null + K i N < 0 , Open image in new window
(4.13)

which shows that C ( 2 n , n ) τ ( p ) < 0 Open image in new window. Therefore τ ( p ) < 0 Open image in new window, which is a proof of the statement (b). □

Notes

Acknowledgements

The authors have greatly benefited from the referee’s report. So we wish to express our gratitude to the reviewer for his/her valuable suggestions which improved the content and presentation of the paper.

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© Gülbahar et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Mehmet Gülbahar
    • 1
  • Erol Kilic̣
    • 1
  • Sadık Keleṣ
    • 1
  1. 1.Department of Mathematics, Faculty of Arts and Sciencesİnönü UniversityMalatyaTurkey

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