Abstract
We define GCR-lightlike submanifolds of indefinite cosymplectic manifolds and give an example. Then, we study mixed geodesic GCR-lightlike submanifolds of indefinite cosymplectic manifolds and obtain some characterization theorems for a GCR-lightlike submanifold to be a GCR-lightlike product.
1. Introduction
To fill the gaps in the general theory of submanifolds, Duggal and Bejancu [1] introduced lightlike (degenerate) geometry of submanifolds. Since the geometry of -submanifolds has potential for applications in mathematical physics, particularly in general relativity, and the geometry of lightlike submanifolds has extensive uses in mathematical physics and relativity, Duggal and Bejancu [1] clubbed these two topics and introduced the theory of -lightlike submanifolds of indefinite Kaehler manifolds and then Duggal and Sahin [2], introduced the theory of -lightlike submanifolds of indefinite Sasakian manifolds, which were further studied by Kumar et al. [3]. But -lightlike submanifolds do not include the complex and real subcases contrary to the classical theory of -submanifolds [4]. Thus, later on, Duggal and Sahin [5] introduced a new class of submanifolds, generalized-Cauchy-Riemann- (GCR-) lightlike submanifolds of indefinite Kaehler manifolds and then of indefinite Sasakian manifolds in [6]. This class of submanifolds acts as an umbrella of invariant, screen real, contact -lightlike subcases and real hypersurfaces. Therefore, the study of -lightlike submanifolds is the topic of main discussion in the present scenario. In [7], the present authors studied totally contact umbilical -lightlike submanifolds of indefinite Sasakian manifolds.
In present paper, after defining -lightlike submanifolds of indefinite cosymplectic manifolds, we study mixed geodesic -lightlike submanifolds of indefinite cosymplectic manifolds. In [8, 9], Kumar et al. obtained some necessary and sufficient conditions for a -lightlike submanifold of indefinite Kaehler and Sasakian manifolds to be a -lightlike product, respectively. Thus, in this paper, we obtain some characterization theorems for a -lightlike submanifold of indefinite cosymplectic manifold to be a -lightlike product.
2. Lightlike Submanifolds
Let be a real -dimensional vector space with a symmetric bilinear mapping . The mapping is called degenerate on if there exists a vector of such that otherwise is called nondegenerate. It is important to note that a non-degenerate symmetric bilinear form on may induce either a non-degenerate or a degenerate symmetric bilinear form on a subspace of . Let be a subspace of and degenerate; then is called a degenerate (lightlike) subspace of .
Let be a real -dimensional semi-Riemannian manifold of constant index such that , , and let be an -dimensional submanifold of and the induced metric of on . Thus, if is degenerate on the tangent bundle of , then is called a lightlike (degenerate) submanifold of (for detail see [1]). For a degenerate metric on , is also a degenerate -dimensional subspace of . Thus, both and are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace , which is known as radical (null) subspace. If the mapping defines a smooth distribution on of rank , then the submanifold of is called an -lightlike submanifold and is called the radical distribution on . Then, there exists a non-degenerate screen distribution which is a complementary vector subbundle to in . Therefore, where denotes orthogonal direct sum. Let , called screen transversal vector bundle, be a non-degenerate complementary vector subbundle to in . Let and be complementary (but not orthogonal) vector bundles to in and to in , called transversal vector bundle and lightlike transversal vector bundle of , respectively. Then, we have
Let be a local coordinate neighborhood of and consider the local quasiorthonormal fields of frames of along on as , where are local lightlike bases of and and are local orthonormal bases of and , respectively. For these quasiorthonormal fields of frames, we have the following theorem.
Theorem 2.1 (see [1]). Let be an -lightlike submanifold of a semi-Riemannian manifold . Then there, exist a complementary vector bundle ltr(TM) of Rad TM in and a basis of consisting of smooth section of , where is a coordinate neighborhood of , such that where is a lightlike basis of .
Let be the Levi-Civita connection on . Then, according to decomposition (2.4), the Gauss and Weingarten formulas are given by for any and , where and belong to and , respectively. Here is a torsion-free linear connection on , is a symmetric bilinear form on that is called second fundamental form, and is a linear operator on , known as shape operator.
According to (2.3), considering the projection morphisms and of on and , respectively, then (2.6) gives where we put , .
As and are -valued and -valued, respectively, they are called the lightlike second fundamental form and the screen second fundamental form on . In particular, where , and . By using (2.3)-(2.4) and (2.7)-(2.8), we obtain for any , , and .
Let be the projection morphism of on . Then, using (2.2), we can induce some new geometric objects on the screen distribution on as for any and , where and belong to and , respectively. and are linear connections on complementary distributions and , respectively. Then, using (2.7), (2.8), and (2.11), we have
Next, an odd-dimensional semi-Riemannian manifold is said to be an indefinite almost contact metric manifold if there exist structure tensors , where is a tensor field, is a vector field called structure vector field, is a -form, and is the semi-Riemannian metric on satisfying (see [10]) for any .
An indefinite almost contact metric manifold is called an indefinite cosymplectic manifold if (see [11])
3. Generalized Cauchy-Riemann Lightlike Submanifolds
Calin [12] proved that if the characteristic vector field is tangent to , then it belongs to . We assume that the characteristic vector is tangent to throughout this paper. Thus, we define the generalized Cauchy-Riemann lightlike submanifolds of an indefinite cosymplectic manifold as follows.
Definition 3.1. Let be a real lightlike submanifold of an indefinite cosymplectic manifold such that the structure vector field is tangent to ; then is called a generalized-Cauchy-Riemann- (GCR-) lightlike submanifold if the following conditions are satisfied:(A)there exist two subbundles and of such that (B)there exist two subbundles and of such that where is invariant nondegenerate distribution on , is one-dimensional distribution spanned by , and and are vector subbundles of and , respectively.
Therefore, the tangent bundle of is decomposed as A contact -lightlike submanifold is said to be proper if , and . Hence, from the definition of -lightlike submanifolds, we have that (a)condition (A) implies that ,(b)condition (B) implies that and , and thus and . (c)any proper -dimensional contact -lightlike submanifold is -lightlike, (d)(a) and contact distribution imply that index .The following proposition shows that the class of -lightlike submanifolds is an umbrella of invariant, contact and contact -lightlike submanifolds.
Proposition 3.2. A -lightlike submanifold of an indefinite cosymplectic manifold is contact -submanifold (resp., contact -lightlike submanifold) if and only if (resp., ).
Proof. Let be a contact -lightlike submanifold; then is a distribution on such that . Therefore, and . Since , this implies that . Conversely, suppose that is a -lightlike submanifold of an indefinite Cosymplectic manifold such that . Then, from (3.1), we have , and therefore . Hence, is a vector subbundle of . This implies that is a contact -lightlike submanifold of an indefinite cosymplectic manifold. Similarly the other assertion follows.
The following construction helps in understanding the example of -lightlike submanifold. Let be with its usual Cosymplectic structure and given by
where are the Cartesian coordinates.
Example 3.3. Let be a semi-Euclidean space and a -dimensional submanifold of that is given by
where is of signature with respect to the canonical basis . Then, the local frame of is given by
Hence, is a -lightlike as . Also, and ; these imply that and , respectively. Since , . By straightforward calculations, we obtain
where ; this implies that . Moreover, the lightlike transversal bundle is spanned by
where and . Hence, . Therefore, . Thus, is a -lightlike submanifold of .
Let , , be the projection morphism on , , , respectively; therefore
for . Applying to (3.9), we obtain
where , , and , or, we can write (3.10) as
where and are the tangential and transversal components of , respectively.
Similarly,
where and are the sections of and , respectively. Differentiating (3.10) and using (2.8)–(2.10) and (3.12), we have
for all . By using, cosymplectic property of with (2.7), we have the following lemmas.
Lemma 3.4. Let be a -lightlike submanifold of an indefinite cosymplectic manifold ; then one has where and
Lemma 3.5. Let be a -lightlike submanifold of an indefinite cosymplectic manifold ; then one has where and and
4. Mixed Geodesic GCR-Lightlike Submanifolds
Definition 4.1. A -lightlike submanifold of an indefinite cosymplectic manifold is called mixed geodesic -lightlike submanifold if its second fundamental form satisfies , for any and .
Definition 4.2. A -lightlike submanifold of an indefinite cosymplectic manifold is called geodesic -lightlike submanifold if its second fundamental form satisfies , for any .
Theorem 4.3. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, is mixed geodesic if and only if and , for any and .
Proof. Using, definition of -lightlike submanifolds, is mixed geodesic if and only if , for , and . Using (2.8) and (2.11), we get Therefore, from (4.1), the proof is complete.
Theorem 4.4. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, is geodesic if and only if and , for any , and .
Proof. The proof is similar to the proof of Theorem 4.3.
Lemma 4.5. Let be a mixed geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Then , for any , .
Proof. For and , using (2.7) we have Since is mixed geodesic, we obtain . Here, using (2.11), we get , and then, by virtue of (3.11), we obtain . Comparing the transversal components, we get ; this implies that If , then the nondegeneracy of implies that there must exist a such that . But using the hypothesis that is a mixed geodesic with (2.7) and (2.11), we get Therefore, Also using (2.13), and (2.15), we get Therefore, Hence, from (4.3), (4.5), and (4.7), the result follows.
Corollary 4.6. Let be a mixed geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Then, , for any and .
Proof. The result follows from (2.12) and Lemma 4.5.
Theorem 4.7. Let be a mixed geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Then, and , for any and .
Proof. Since is mixed geodesic -lightlike submanifold for any , and thus (2.6) implies that Since is an anti-invariant distribution there exists a vector field such that . Thus, from (2.8), (2.14), (3.11), and (3.12), we get Comparing the transversal components, we get . Since and , this implies that and . Hence, and .
5. GCR-Lightlike Product
Definition 5.1. -lightlike submanifold of an indefinite cosymplectic manifold is called -lightlike product if both the distributions and define totally geodesic foliation in .
Theorem 5.2. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, the distribution define a totally geodesic foliation in if and only if , for any .
Proof. Since , defines a totally geodesic foliation in if and only if , for any , , and . Using (2.7) and (2.14), we have Hence, from (5.1) and (5.2), the assertion follows.
Theorem 5.3. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, the distribution defines a totally geodesic foliation in if and only if has no component in and has no component in , for any and .
Proof. From the definition of a -lightlike submanifold, we know that defines a totally geodesic foliation in if and only if for and . Using (2.7) and (2.8), we have Using (2.7), (2.15), and (2.14), we obtain Thus, from (5.4)–(5.7), the result follows.
Theorem 5.4. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . If , then is a lightlike product.
Proof. Let ; therefore . Then using (3.15) with the hypothesis, we get . Therefore the distribution defines a totally geodesic foliation. Next, let ; therefore . Then using (3.14), we get . Therefore, defines a totally geodesic foliation in . Hence, is a lightlike product.
Definition 5.5. A lightlike submanifold of a semi-Riemannian manifold is said to be an irrotational submanifold if , for any and . Thus, is an irrotational lightlike submanifold if and only if and .
Theorem 5.6. Let be an irrotational -lightlike submanifold of an indefinite cosymplectic manifold . Then, is a lightlike product if the following conditions are satisfied:(A), (B).
Proof. Let hold; then, using (2.8), we get , , and for . These equations imply that the distribution defines a totally geodesic foliation in , and, with (2.9), we get . Hence, the non degeneracy of implies that . Therefore, has no component in . Finally, from (2.10) and the hypothesis that is irrotational, we have , for and . Assume that holds; then . Therefore, has no component in . Thus, the distribution defines a totally geodesic foliation in . Hence, is a lightlike product.
Definition 5.7 (see [13]). If the second fundamental form of a submanifold, tangent to characteristic vector field , of a Sasakian manifold is of the form for any , where is a vector field transversal to , then is called a totally contact umbilical submanifold of a Sasakian manifold.
Theorem 5.8. Let be a totally contact umbilical -lightlike submanifold of an indefinite cosymplectic manifold . Then, is a -lightlike product if , for any .
Proof. Let ; then the hypothesis that implies that the distribution defines a totally geodesic foliation in .
If we assume that , then, using (3.14), we have , and taking inner product with and using (2.6) and (2.14), we obtain
where . For any from (3.14), we have . Therefore, using the hypothesis with (5.8), we get ; this implies that , and thus (5.9) becomes . Then, the nondegeneracy of the distribution implies that the distribution defines a totally geodesic foliation in . Hence, the assertion follows.
Theorem 5.9. Let be a totally geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Suppose that there exists a transversal vector bundle of which is parallel along with respect to Levi-Civita connection on , that is, , for any , . Then, is a -lightlike product.
Proof. Since is a totally geodesic -lightlike , for ; this implies defines a totally geodesic foliation in .
Next implies , and hence, by Theorem 5.3, the distribution defines a totally geodesic foliation in . Hence, the result follows.
Acknowledgment
The authors would like to thank the anonymous referee for his/her comments that helped them to improve this paper.