GCR-Lightlike Product of Indefinite Kaehler Manifolds

We study geodesic 𝐺𝐶𝑅-lightlike submanifolds of indefinite Kaehler manifolds and obtain some necessary and sufficient conditions for a 𝐺𝐶𝑅-lightlike submanifold to be a 𝐺𝐶𝑅-lightlike product.


Introduction
The geometry of CR-submanifolds of Kaehler manifolds was initiated by Bejancu 1 , which includes holomorphic and totally real submanifolds as subcases, and further developed by Bejancu 2 ,Bejancu et al. 3 , Blair and Chen 4 , Chen 5 , Yano and Kon 6, 7 , and many others.They all studied the geometry of CR-submanifolds with positive definite metric.Therefore this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite.Thus the geometry of CR-submanifolds with indefinite metric became a topic of chief discussion and Duggal 8, 9 played a very crucial role.Duggal and Bejancu 10 introduced the notion of CR-lightlike submanifolds which exclude the totally real and complex subcases.Then Duggal and Sahin 11 introduced SCR-lightlike submanifolds which contain complex and totally real subcases but there was no inclusion relation between CR and SCR-cases.Thus to find a class of submanifolds which would behave as an umbrella for CR-lightlike and SCR-lightlike submanifolds of an indefinite Kaehler manifold, Duggal and Sahin 12 introduced GCR-lightlike submanifolds of indefinite Kaehler manifolds.This paper starts with a very brief introduction about lightlike geometry and GCR-lightlike submanifolds which will be needed throught the paper and then we study geodesic GCR-lightlike submanifolds and obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product.

Lightlike Submanifolds
We recall notations and fundamental equations for lightlike submanifolds, which are due to the book 8 by Duggal and Bejancu.
Let M, g be a real m n -dimensional semi-Riemannian manifold of constant index q such that m, n ≥ 1, 1 ≤ q ≤ m n − 1 and M, g is an m-dimensional submanifold of M and g is the induced metric of g on M. If g is degenerate on the tangent bundle TM of M, then M is called a lightlike submanifold of M. For a degenerate metric g on M Let ∇ be the Levi-Civita connection on M; then according to the decomposition 2.5 , the Gauss and Weingarten formulas are given by where {∇ X Y, A U X} and {h X, Y , ∇ ⊥ X U} belong to Γ TM and Γ tr TM , respectively.Here ∇ is a torsion-free linear connection on M, h is a symmetric bilinear form on Γ TM which is called second fundamental form, and A U is a linear a operator on M and known as shape operator.
According to 2.4 considering the projection morphisms L and S of tr TM on ltr TM and S TM ⊥ , respectively, then 2.7 and 2.8 become where we put As h l and h s are Γ ltr TM -valued and Γ S TM ⊥ -valued, respectively, therefore they are called the lightlike second fundamental form and the screen second fundamental form on M. In particular where X ∈ Γ TM , N ∈ Γ ltr TM , and W ∈ Γ S TM ⊥ .Using 2.9 -2.12 we obtain for any ξ ∈ Γ Rad TM , W ∈ Γ S TM ⊥ , and N, N ∈ Γ ltr TM .

ISRN Geometry
Let P be the projection morphism of TM on S TM ; then using 2.3 , we can induce some new geometric objects on the screen distribution S TM on M as 16 From the geometry of Riemannian submanifolds and nondegenerate submanifolds, it is known that the induced connection ∇ on a nondegenerate submanifold is a metric connection.Unfortunately, this is not true for lightlike submanifolds.Indeed considering ∇ a metric connection, we have for any X, Y, Z ∈ Γ TM .From 8, page 171 , using the properties of linear connection, we have

2.19
Barros and Romero 13 defined indefinite Kaehler manifolds as follows.
Definition 2.2.Let M, J, g be an indefinite almost Hermitian manifold and let ∇ be the Levi-Civita connection on M with respect to g.Then M is called an indefinite Kaehler manifold if J is parallel with respect to ∇, that is,

Generalized Cauchy-Riemann Lightlike Submanifolds
Definition 3.1.Let M, g, S TM be a real lightlike submanifold of an indefinite Kaehler manifold M, g, J , then M is called a generalized Cauchy-Riemann GCR -lightlike submanifold if the following conditions are satisfied.
A There exist two subbundles D 1 and D 2 of Rad TM such that B There exist two subbundles D 0 and D of S TM such that where D 0 is a nondegenerate distribution on M, and L 1 and L 2 are vector bundle of ltr TM and S TM ⊥ , respectively.
Then the tangent bundle TM of M is decomposed as where TX and wX are the tangential and transversal components of JX, respectively.Similarly JV BV CV, 3.7 for any V ∈ Γ tr TM , where BV and CV are the sections of TM and tr TM , respectively.Differentiating 3.5 and using 2.9 -2.12 and 3.7 we have

3.8
Using Kaehlerian property of ∇ with 2.11 and 2.12 , we have the following lemmas.
Lemma 3.2.Let M be a GCR-lightlike submanifold of an indefinite Kaehlerian manifold M. Then one has where X, Y ∈ Γ TM and

3.12
Lemma 3.3.Let M be a GCR-lightlike submanifold of an indefinite Kaehlerian manifold M. Then one has

Geodesic GCR-Lightlike Submanifolds
for any ξ JY ∈ Γ Rad TM .Now using 4.4 and 4.5 in 4.3 , we obtain Hence the second part of the assertion follows from 4.6 .

Theorem 4.5. Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then M is D -geodesic if and only if A W X and A
Proof.For X, Y ∈ Γ D and W ∈ Γ S TM ⊥ using 2.13 , we obtain and for ξ ∈ Γ Rad TM using 2.14 and 2.17 we obtain Hence the assertion follows from 4.7 and 4.8 .
Theorem 4.6.Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then M is mixed geodesic if and only if for any X ∈ Γ D , ξ ∈ Γ Rad TM , and W ∈ Γ S TM ⊥ .
Proof.For any X ∈ Γ D , Y ∈ Γ D , and ξ ∈ Γ Rad TM using 2.14 and 2.17 we obtain and for W ∈ Γ S TM ⊥ with 2.13 , we obtain Hence the result follows from 4.10 and 4.11 .
Theorem 4.7.Let M be a mixed geodesic GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then one has

4.13
Since M is mixed geodesic, therefore Using 2.16 and 2.17 we obtain

4.15
Equating the transversal components we have wA * ξ X 0.

4.16
Thus Now, for Z ∈ Γ D 0 and ξ ∈ Γ D 2 we have

4.18
If A * ξ X ∈ Γ D 0 , then using the nondegeneracy of D 0 for any Z ∈ Γ D 0 , we have g A * ξ X, Z / 0. Therefore A * ξ X / ∈ Γ D 0 .Hence the assertion is proved.
Theorem 4.8.Let M be a mixed geodesic GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then the transversal section V ∈ Γ JD is D-parallel if and only if ∇ X JV ∈ Γ D , for any X ∈ Γ D .

ISRN Geometry
Proof.Let Y ∈ Γ D such that JY wY V ∈ Γ L 1 ⊥ L 2 and X ∈ Γ D ; then using hypothesis in 3.9 we have T ∇ Since ∇ is a Kaehlerian connection and M is mixed geodesic, therefore we have ∇ t X V w∇ X Y or consequently ∇ t X V −w∇ X JV , which clearly proves the theorem.
Theorem 4.9.Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold M such that Proof.For X ∈ Γ D , Y ∈ Γ D , and V ∈ Γ L ⊥ 1 , we have

4.22
Hence the assertion follows from 4.19 -4.22 .Proof.For any X, Y ∈ Γ D , 3.9 implies that T ∇ X Y −A wY X − Bh X, Y ; then for Z ∈ Γ D 0 we have

GCR-Lightlike Product
where Z JZ ∈ Γ D 0 .Since X ∈ Γ D and Z ∈ Γ D 0 , then from 3.8 we have wP ∇ X Z h X, T Z − Ch X, Z Hg X, T Z − CHg X, Z 0, therefore wP ∇ X Z 0, and this implies that ∇ X Z ∈ Γ D .Therefore 5.1 implies that g T ∇ X Y, Z 0; then the nondegeneracy of D 0 implies that T ∇ X Y 0. Hence ∇ X Y ∈ Γ D , for any X, Y ∈ Γ D .Thus the result follows.Conversely, let M be a GCR-lightlike product; therefore the distribution D defines a totally geodesic foliation in M. Using Kaehlerian property of ∇, for any X, Y ∈ Γ D we have ∇ X JY J ∇ X Y ; then comparing transversal components, we obtain h X, JY Jh X, Y and then ∇ X T Y ∇ X TY − T ∇ X Y ∇ X JY − h X, JY − J ∇ X Y h X, JY 0, that is, ∇ X T Y 0, for any X, Y ∈ Γ D .Let D defines a totally geodesic foliation in M, and using Kaehlerian property of ∇, we have ∇ X JY J ∇ X Y ; then comparing tangential components on both sides, we obtain −A wY X Bh X, Y ; then 3.9 implies that ∇ X T Y 0, which completes the proof.
for any X, Y ∈ Γ TM and ξ ∈ Γ Rad TM , where {∇ * X PY, A * ξ X} and {h * X, Y , ∇ * t X ξ} belong to Γ S TM and Γ Rad TM , respectively.∇ * and ∇ * t are linear connections on complementary distributions S TM and Rad TM, respectively.h * and A * are Γ Rad TMvalued and Γ S TM -valued bilinear forms and are called as second fundamental forms of distributions S TM and Rad TM, respectively.

Theorem 3 . 4
12 investigated the conditions to define totally geodesic foliations by the distributions D and D in M as follows.see 12 .Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then the distribution D defines a totally geodesic foliation in M if and only if Bh X, Y 0, for any X, Y ∈ Γ D .Theorem 3.5 see 12 .Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then the distribution D defines a totally geodesic foliation in M if and only if A wY X ∈ Γ D , for any X, Y ∈ Γ D .

Definition 4 . 1 .
A GCR-lightlike submanifold of an indefinite Kaehler manifold is called mixed geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h X, Y 0 for any X ∈ Γ D and Y ∈ Γ D .Definition 4.2.A GCR-lightlike submanifold of an indefinite Kaehler manifold is called D geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h X, Y 0 for any X, Y ∈ Γ D .Definition 4.3.A GCR-lightlike submanifold of an indefinite Kaehler manifold is called D geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h X, Y 0 for any X, Y ∈ Γ D .

Theorem 4 . 4 .
Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then M is D-geodesic if and only if

Definition 5 . 1 .Lemma 5 . 2 .
A GCR-lightlike submanifold M of an indefinite Kaehler manifold M is called a GCR-lightlike product if both the distributions D and D define totally geodesic foliations in M. Let M be a totally umbilical GCR-lightlike submanifold of an indefinite Kaehler manifold M; then the distribution D defines a totally geodesic foliation in M.

Theorem 5 . 3 .
Let M be a totally umbilical GCR-lightlike submanifold of an indefinite Kaehler manifold M. Then M is a GCR-lightlike product if and only if Bh X, Y 0, for any X ∈ Γ TM and Y ∈ Γ D .Proof.Let M be a GCR-lightlike product; therefore the distributions D and D define a totally geodesic foliation in M. Therefore using Theorem 3.4, Bh X, Y 0 for any X, Y ∈ Γ D .Now let X ∈ Γ D and Y ∈ Γ D ; then Bh X, Y g X, Y BH 0. Hence Bh X, Y 0, for any X ∈ Γ TM and Y ∈ Γ D .
Thus both T x M and T x M ⊥ are degenerate orthogonal subspaces but no longer complementary.In this case, there exists a subspace Rad T Let u be a local coordinate neighborhood of M and consider the local quasiorthonormal fields of frames of M along M, on u as {ξ 1 , . . ., ξ r , W r 1 , . . ., W n , N 1 , . . ., N r , X r 1 , . . ., X m }, where {ξ 1 , . . ., ξ r } and {N 1 , . . ., N r } are local lightlike bases of Γ Rad TM| u and Γ ltr TM | u , and {W r 1 , . . ., W n } and {X r 1 , . . ., X m } are local orthonormal bases of Γ S TM ⊥ | u and Γ S TM | u , respectively.For this quasiorthonormal fields of frames, we have the following. .Then there exist a complementary vector bundle ltr TM of Rad TM in S TM ⊥ ⊥ and a basis of Γ ltr TM | u consisting of smooth section {N i } of S TM ⊥ ⊥ | u , where u x M T x M ∩ T x M ⊥ which is known as radical null subspace.If the mapping Rad TM : x ∈ M −→ Rad T x M 2.2 defines a smooth distribution on M of rank r > 0, then the submanifold M of M is called an r-lightlike submanifold and Rad TM is called the radical distribution on M. Screen distribution S TM is a semi-Riemannian complementary distribution of Rad TM in TM, that is, TM Rad TM ⊥ S TM , 2.3 and S TM ⊥ is a complementary vector subbundle to Rad TM in TM ⊥ .Let tr TM and ltr TM be complementary but not orthogonal vector bundles to TM in T M| M and to Rad TM in S TM ⊥ ⊥ , respectively.
and L 2 / {0}.Let Q, P 1 , and P 2 be the projections on D, J L 1 M 1 and J L 2 M 2 , respectively.Then for any X ∈ Γ TM we have and W ∈ Γ S TM ⊥ .Proof.Using the definition of GCR-lightlike submanifolds, M is D-geodesic, if and only if for any X, Y ∈ Γ D , ξ ∈ Γ Rad TM , and W ∈ Γ S TM ⊥ .Thus for X, Y ∈ Γ D , first part of the assertion follows from 2.13 .Now for X, Y ∈ Γ D , ξ ∈ Γ Rad TM using 2.16 , we have