1. Introduction
The Riemann curvature tensor is one of the central concepts in the mathematical field of differential geometry. It assigns a tensor to each point of a (semi-)Riemannian manifold that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. It expresses the curvature of (semi-)Riemannian. Curvature tensor is a central mathematical tool in the theory of general relativity and gravity.
The geometry of a pseudo-Riemannian manifold is the study of the curvature which is defined by the Levi-Civita connection . Since the whole curvature tensor is difficult to handle, the investigation usually focuses on different objects whose properties allow us to recover curvature tensor. One can for example associate to R an endomorphism on tangent bundle of a manifold. In [1] P. Gilkey studied geometric properties of natural operators defined by the Riemann curvature tensor and Osserman proposed in [2] a characterization of Riemannian rank 1-symmetric spaces in terms of the spectrum of the Jacobi operator. Many other central works have been done by Ivanova, Stanilov, Videv and Szabo [3] [4] [5] [6] [7].
On lightlike geometry of hypersurfaces, C. Atindogbe and K. L. Duggal have studied Pseudo-Jacobi operators and considered Osserman conditions [8], and in [9], the authors introduced the notion of r-lightlike Osserman submanifolds.
Let be a semi-Riemannian manifold and . An element is said to be an algebraic curvature tensor on if R has the following symmetries:
(1)
(2)
The Riemannian curvature tensor of a Levi-Civita connection is algebraic on for all . If R is an algebraic curvature tensor on , the associated Jacobi operator with respect to is the self-adjoint linear map on characterized by the identity
(3)
It is obvious that and the domain of is the unit pseudo-sphere of unit timelike or unit spacelike vectors
Due to the algebraic properties (1) and (2) of the curvature, we have and . Then, the Jacobi operator naturally reduces to the endomorphism .
The Riemannian curvature tensor R of a semi-Riemannian manifold is said to be a spacelike (resp. timelike) Osserman tensor on if the spectrum spec ( ) is constant on (resp. ). If this is the case at each , we say that is pointwise Osserman semi-Riemannian manifold.
Motivated by the recent works on lightlike geometry, we consider in this paper lightlike warped product (sub-)manifolds and examine Osserman conditions depending on geometric properties of the factors.
In Section 2, we present background materials of lightlike geometry. In Section 3 we define lightlike warped product Osserman (definition 3.2) and present some important results of our research (Theorem 2, Theorem 3, Theorem 4). Section 4 is concerned with an example given in the neutral semi-Riemannian space ..
2. Preliminaries
Let be a -dimensional semi-Riemannian manifold of constant index q such that and be an m-dimensional submanifold of . We assume that both m and k are . At each point ,
(4)
is the normal space at p. In case is non-degenerate on , both and are non-degenerate and we have . If the mapping
(5)
is a smooth distribution with constant rank , M is said to be lightlike (or null) submanifold of , with nullity degree r. This mapping is called the radical distribution on M. Any complementary (and hence orthogonal) distribution of in TM is called a screen distribution. For a fixed screen distribution on M, the tangent bundle splits as
(6)
where is the orthogonal direct sum.
A screen transversal vector bundle on M is any (semi-Riemannian) complementary vector bundle of in . It is obvious that both and is non-degenerate with respect to and
(7)
A null submanifold M with nullity degree r equipped with a screen distribution and a screen transversal vector bundle is denoted . It is said to be
• r-lightlike if ;
• coisotropic if (hence );
• isotropic if , (hence );
• totally null if , (hence ).
For any local frame of , there exists a local frame of sections with values in the orthogonal complement of in such that
and it follows that there exists a lightlike transversal vector bundle locally spanned by .
If we denote by a (not orthogonal) complementary vector bundle to in , the following relations hold
(8)
(9)
The Gauss and Weingarten formulas are
(10)
(11)
. The components and belong to , and to . and are linear connections on TM and the vector bundle respectively. According to the decomposition (8), let L and S denote the projection morphisms of onto and respectively, , where is the composition law, , . The transformations and do not define linear connections but Otsuki connections on with respect to the vector bundle morphisms L and S. Then,
(12)
(13)
(14)
and .
Since is a metric connection, using (12)-(14) we have
(15)
(16)
Let P the projection morphism of TM onto . Using the decomposition (6) we get
(17)
(18)
and is a metric connection on .
It follows from (17) and (18) that
(19)
(20)
(21)
Let and R denote the Riemannian curvature tensors on and M respectively. The Gauss equation is given by
(22)
. Therefore
(23)
Definition 2.1 ( [10]). A lightlike submanifold of a semi-Riemannian manifold is totally umbilical in if there is a smooth transversal vector field on M called the transversal curvature vector field of M such that, for all
(24)
Using (10) and (12) it is easy to see that M is totally umbilical if and only if on each coordinate neighbourhood there exist smooth vector fields and such that
(25)
Definition 2.2 ( [10]). Let be a r-lightlike (i.e. ) or a coisotropic m-dimensional submanifold of a -dimensional semi-Riemannian manifold . We say that the screen distribution is totally umbilical if for any section N of on a coordinate neighbourhood , there exists a smooth function on such that
(26)
Definition 2.3. A coisotropic submanifold of a semi-Riemannian manifold is screen locally conformal if the local second fundamental forms of the screen distribution are related with the local second fundamental form of M as follows:
(27)
where is a conformal smooth function on a coordinate neighbourhood in M. In particular, we say that M is screen homothetic if is a non-zero constant.
Let be a null submanifold with nullity degree r of a semi-Riemannian manifold , and local frames of and respectively satisfying . Consider the 1-forms metrically equivalent to the i.e. . Then, each tangent vector field X has the splitting,
(28)
From now on, we assume that the frames and are globally defined on M. Consider the values mapping defined on by
(29)
where
denotes the interior product with respect to X. The mapping
is an isomorphisme of
onto
and we let
denote its reverse mapping. (resp
),
is called the dual 1-form of X and
the dual field of
with respect to the pair of frames
and
. Define a
-tensor
on M by
(30)
i.e.
It is straightforward to check that defines a non-degenerate metric on M and that for
it coincides with g. The
-tensor
is called the pseudo-inverse of g. Let
be a quasi-orthonormal field of frames on M with respect to the decomposition (6). Using (30) we have
Definition 2.4. Let and
be semi-Riemannian and
be positive smooth functions. The multiply warped product
is the product manifold
furnished with the metric tensor
where,
are the projection morphisms. The functions
are called the warping functions and
the base manifold of the multiply warped product. Each
is called a fiber manifold.
• If then we obtain a singly warped product
• If for
then we have a multiple product manifold.
• If all are Riemanniann then
is also a Riemannian multiply warped product manifold.
is Lorentzian multiply warped product if
are Riemannian and either
is Lorentzian or a one-dimensional manifold with a negative definite metric
.
• is lightlike with nullity degree r if
is degenerate with
of rank r.
still has rank r and all screen structure on M has dimension
where
is the dimension of any screen structure on
.
Proposition 1. [11] On, if
;
, then
1) is the lift of
,
2),
3) is the lift of
,
4).
Corollary 1. The leaves of the warped product are totally geodesic; the fibers
are totally umbilical.
3. Lightlike Warped Product Geometry and Osserman Conditions
As it is well known, Jacobi operators are associated to algebraic curvature maps (tensors). But contrary to non-lightlike manifolds, the induced Riemann curvature tensor of a lightlike submanifold is not an algebraic curvature map in general as it can be seen from (23). In case this requirement is satisfied, the pair of screens
is said to be admissible.
In semi-Riemannian case, the relation (3) characterizes the Jacobi operator associated to an algebraic curvature tensor
. For
(or
),
, we have
(31)
that is
(32)
For degenerate warped product setting, we consider the associated non-degenerate metric defined by (30) of a lightlike warped product metric. We denote by
and
the natural isomorphisms with respect to
. The equivalent relation of (3) is given by
(33)
Definition 3.1. Let be a lightlike warped product submanifold of a semi-Riemannian manifold
,
,
(or
) and
an algebraic curvature tensor on
. A pseudo-Jacobi operator associated to R with respect to X is the self-adjoint linear map
on
defined by
(34)
or equivalentently
(35)
Definition 3.2. A lightlike warped product submanifold of a semi-Riemannian manifold
is called spacelike (resp. timelike) Osserman at
if for each admissible pair of screens
and associate induced Riemann curvature R, the characteristic polynomial of
is independent of
(resp.
). If this is the case for each
, then
is called pointwise Osserman. If in addition there is no dependence with respect to
then
is said to be globally Osserman.
Theorem 2. Let and
be a totally lightlike manifold and a conformally Osserman semi-Riemannian manifold respectively. Let f be an isometric immersion of
in a semi-Riemannian space form
where
and
. Then
is a conformally Osserman Lightlike warped product submanifold of
.
Proof. Let and R be the Riemannian curvature tensors of
and N respectively.
being totally degenerate, the Riemannian curvature tensor
and its Weyl tensor vanish identically. Moreover
is conformally Osserman. By Theorem 5 in [12], R is an algebraic curvature tensor. If we restrict our study on the product
, it is obvious that N is a conformally Osserman manifold. The lightlike warped product metric g belongs
to the conformal class of which is conformally Osserman lightlike
product submanifold. Since the Weyl tensor is invariant in the conformal class of a metric, we conclude that is a conformally Osserman lightlike warped product.
From definition 2.2, it is obvious that if a screen distribution is totally umbilical, the bilinear form
is symmetric on
. By theorem 2.5 in [13] p.161,
is integrable and by theorem 5.3 in [10], if the ambiant space is of constant sectional curvature, the induced Ricci tensor is symmetric.
Due to Proposition 2 in [12], we establish the following two results for coisotropic warped product with
totally degenerate isometrically immersed in a semi-Riemannian space form
.
Theorem 3. Let be a coisotropic isometric immersion of a warped product of a totally lightlike manifold
and a semi-Riemannian manifold
in a semi-Riemannian space form
that is screen conformal. Then the associated Ricci tensor is symetric and N is locally Einstein. Also, N is pointwise Osserman.
Proof. From (22), the induced Riemannian curvature tensor is
(36)
Using (20) and (27), for all, we get
(37)
Let be a quasi-orthonormal field of frames
on where
is a orthonormal field of frames on
. Then, for all
we compute the induced Ricci curvator tensor as follow
Thus the induced Ricci curvature tensor is symmetric and N is locally Einstein. Consider and
. By (37) we have
The pseudo-Jacobi operator is given by
and its characteristic polynomial is
Therefore N is pointwise Osserman. ■
From Proposition 2, theorem 5 in [12] and Theorem 4.3 in [9], we proved the following result that characterizes any screen distribution of a coisotropic warped product of a semi-Riemannian space form with the first factor totally null. This case consists of a class of null warped products that is Einstein and pointwise Osserman.
Theorem 4. Let, be a coisotropic warped product submanifold of a semi-Riemannian space forme
where
is totally degenerate. Then any screen distribution is admissible and totally umbilical on N. In addition N is locally Einstein and pointwise Osserman.
4. Example
Let be a semi-Riemannian manifold, where
is semi-Euclidean space of signature
with respect to the canonical basis
. Let M be submanifold of
given by
where.
Then is spanned by
. Thus M is a 2-lightlike submanifold of
with
. Choose
, we construct two null vectors
Since rank equals codimension of M, we conclude that M is a coisotropic submanifold of
. By direct calculations,
and
are integrable in M. The induced metric tensor
on M is given by
and we get
It is obvious that M is a coisotropic warped product submanifold of with warping function
. Using (12) and (17) we obtain
(38)
(39)
From (38), we have
(40)
and we conclude that is a totally umbilical null warped product submanifold of
and it is obvious that it is mixed totally geodesic.
From (39) we have
(41)
(42)
and we conclude that the screen distribution of M is totally umbilical and M is screen homothetic.
M being coisotropic, taking into account has constant sectional curvature
, consider (40) and (41), from (42) we get
(43)
Let’s consider, (or
),
. Then by using (43), we have
The pseudo-Jacobi operator is given by
and its characteristic polynomial
is given by
(44)
that is independent of. Therefore, M is spacelike (timelike) pointwise Osserman null warped product.
Remark. From (44), it is obvious that for a lightlike warped product manifold, to be spacelike Osserman or timelike Osserman are equivalent.
5. Conclusion and Suggestions
Osserman conditions on lightlike warped product manifolds have been considered in this paper. The case of lightlike warped product with the first factor totally degenerate has been explored. Especially in coisotropic case, we have proved that this class consists of Einstein and locally Osserman lightlike warped product. In perspective, we are going to extend this study to other classes of lightlike warped product in order to get later a certain characterization of lightlike warped product Osserman manifolds.
[1] Gilkey, P. (2002) Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor. World Scientific, Singapore.
https://doi.org/10.1142/4812
[2] Osserman, R. (1990) Curvature in the Eighties. The American Mathematical Monthly, 97, 731-756.
https://doi.org/10.1080/00029890.1990.11995659
[3] Ivanova, R. and Stanilov, G. (1994) A Skew-Symmetric Curvature Operator in Riemannian Geometry. Symposia Gaussiana. Conference A: Mathematics and Theoretical Physics, 391-396.
https://doi.org/10.1515/9783110886726.391
[4] Stanilov, G. (2004) Higher Order Skew-Symmetric and Symmetric Curvature Operators. Comptes rendus de l’Academie bulgare des Sciences, 57, 9-12.
[5] Stanilov, G. and Videv, V. (1992) On a Generalization of the Jacobi Operator in the Riemannian Geometry. Annual of Sofia University “St Kliment Ohridski”, Facurity of Mathematics and Informormatics, 86, 27-34.
[6] Stanilov, G. and Videv, V. (2004) On the Commuting of Curvature Operators. Mathematics and Education in Mathematics, Proceedings of the 33rd Spring Conference of the Union of Bulgarian Mathematicians, Sofia, 1-4 April 2004, 176-179.
[7] Szabo, Z.I. (1991) A Simple Topological Proof for the Symmetry of 2 Point Homogeneous Spaces. Inventiones Mathematicae, 106, 61-64.
https://doi.org/10.1007/BF01243903
[8] Atindogbe, C. and Duggal, K.L. (2009) Pseudo-Jacobi Operators and Osserman Lightlike Hypersurfaces. Kodai Mathematical Journal, 32, 91-108.
[9] Atindogbe, C., Lungiambudila, O. and Tossa, J. (2011) Ligthlike Osserman Submanifolds of Semi-Riemannian Manifolds. Afrika Matematika, 22, 129-151.
[10] Duggal, K.L. and Jin, D.H. (2003) Totally Umbilical Lightlike Submanifolds. Kodai Mathematical Journal, 26, 49-68.
https://doi.org/10.2996/kmj/1050496648
[11] O’Neill, B. (1983) Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York.
[12] Ndayirukiye, D., Nibarura, G., Karimumuryango, M. and Nibirantiza, A. (2019) Algebraicity of Induced Riemannian Curvature Tensor on Lightlike Warped Product Manifolds. Journal of Applied Mathematics and Physics, 7, 3132-3139.
[13] Duggal, K.L. and Bejancu, A. (1996) Lightlike Sub of Semi-Riemannian and Applications. Kluwer Academic Publishers, Amsterdam, 308 p.
https://doi.org/10.1007/978-94-017-2089-2