Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold
© Gülbahar et al.; licensee Springer 2013
Received: 31 December 2012
Accepted: 13 May 2013
Published: 28 May 2013
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 -spaces for lightlike hypersurfaces of a Lorentzian manifold.
where is a shape operator of M, and 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 in  as follows:
where is the squared mean curvature and is Ricci curvature of at X.
where is scalar curvature of M, is j-scalar curvature, run over all k mutually orthogonal subspaces of such that , . In , the authors gave optimal relationships among invariant , 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 [11–13]. Contact versions of Chen inequalities and their applications were introduced in [7, 14–16]. In , 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 -dimensional Ricci-flat space in . The authors in  proved an inequality using the extrinsic and the intrinsic scalar curvature in a semi-Riemannian manifold. In , 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 on lightlike hypersurfaces of a Lorentzian manifold.
where B and 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 .
where . M is called totally umbilical in if every point of M is umbilical .
where and belong to . Here , C and are called the induced connection, the local second fundamental form and the local shape operator on , respectively.
where φ is a non-vanishing smooth function on a neighborhood U on M . In particular, M is called screen homothetic if φ is a non-zero constant.
for any .
is called the sectional curvature at . Since the screen second fundamental form C is not symmetric, the sectional curvature of a lightlike submanifold is not symmetric, that is, .
where k is a constant, then M is called an Einstein hypersurface .
where for .
3 k-Ricci curvature and k-scalar curvature
Let M be an -dimensional lightlike hypersurface of a Lorentzian manifold and let be a basis of where is an orthonormal basis of . For , we set is a -dimensional degenerate plane section and is a k-dimensional non-degenerate plane section.
where and for .
The equalities case of (3.10) and (3.11) hold at if and only if p is a totally geodesic point.
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. □
The equality of (3.19) holds for all if and only if M is minimal.
which implies (3.19).
This shows that M is minimal. □
By Theorem 3.4 we get the following corollaries.
where is equal to (3.20). The equality of (3.23) holds for all if and only if M is minimal.
The equality of (3.24) holds for all if and only if M is minimal.
Now, we shall need the following lemma.
Lemma 3.7 
with equality if and only if .
where is equal to (3.20).
The equality case of (3.26) holds for all if and only if .
which implies (3.26).
Since , . □
From Theorem 3.8 we get the following corollaries.
where is equal to (3.20).
The equality case of (3.31) holds for all if and only if .
The equality case of (3.32) holds for all if and only if either or M is minimal.
4 Curvature invariants on lightlike hypersurfaces
Definition 4.1 For an integer , let be the finite set which consists of k-tuples of integers ≥2 satisfying and . Denote by the set of all unordered k-tuples with for a fixed positive integer n.
We call a lightlike hypersurface an space if it satisfies .
Theorem 4.2 Let M be a lightlike hypersurface of an -dimensional Lorentzian manifold . Then M is an space if and only if the scalar curvature of M is constant.
which shows that is constant, which completes the proof. □
Remark 4.3 We note that if an n-dimensional non-degenerate manifold is an space, then it is an Einstein space (see ). On the other hand, if a degenerate hypersurface of a lightlike hypersurface is an 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 be an n-dimensional lightlike hypersurface with constant sectional curvature c. is an space if and only if .
Now, we prove the following.
Theorem 4.5 Let M be a lightlike hypersurface of an -dimensional Lorentzian manifold . If M is an space for , then M is also space.
Therefore, M is an space.
Now, we show that the claim of the theorem is true for .
From (4.4) and (4.5) M is an space. □
Theorem 4.6 Let M be a lightlike hypersurface of an -dimensional Lorentzian manifold . Let be an orthonormal basis of . If M is an space, then and .
where and . Using Theorem 4.2 and Theorem 4.5, we obtain . In addition to this, from (4.1), Theorem 4.2 and Theorem 4.5, we have , which completes the proof of the theorem. □
admits a canonical screen distribution that induces a canonical lightlike transversal vector bundle ,
admits an induced symmetric Ricci tensor, denoted by Ric0.
From above information, we get the following theorem immediately.
where is any -dimensional null section of and denotes the orthogonal complement in .
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.
If , then M is an -space.
- (b)If , then M is not necessary an -space. If
then M is an -space.
where is a three-dimensional null plane section of .
- (b)We show that the claim of the theorem is true for . Let us choose any three-dimensional plane section of as , , . If M is an -space, then(4.9)
where . 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 .
If , then .
If , then .
which shows that . Therefore, which is a proof of the statement (a).
which shows that . Therefore , which is a proof of the statement (b). □
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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