Abstract
We study geodesic -lightlike submanifolds of indefinite Kaehler manifolds and obtain some necessary and sufficient conditions for a -lightlike submanifold to be a -lightlike product.
1. Introduction
The geometry of -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 -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 -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 -lightlike submanifolds which exclude the totally real and complex subcases. Then Duggal and Sahin [11] introduced -lightlike submanifolds which contain complex and totally real subcases but there was no inclusion relation between and -cases. Thus to find a class of submanifolds which would behave as an umbrella for -lightlike and -lightlike submanifolds of an indefinite Kaehler manifold, Duggal and Sahin [12] introduced -lightlike submanifolds of indefinite Kaehler manifolds. This paper starts with a very brief introduction about lightlike geometry and -lightlike submanifolds which will be needed throught the paper and then we study geodesic -lightlike submanifolds and obtain some necessary and sufficient conditions for a -lightlike submanifold to be a -lightlike product.
2. Lightlike Submanifolds
We recall notations and fundamental equations for lightlike submanifolds, which are due to the book [8] by Duggal and Bejancu.
Let be a real -dimensional semi-Riemannian manifold of constant index such that , and is an -dimensional submanifold of and is the induced metric of on . If is degenerate on the tangent bundle of , then is called a lightlike submanifold of . For a degenerate metric on is 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 .
Screen distribution is a semi-Riemannian complementary distribution of in , that is, and is a complementary vector subbundle to in . Let and be complementary (but not orthogonal) vector bundles to in and to in , respectively. Then we have
Let be a local coordinate neighborhood of and consider the local quasiorthonormal fields of frames of along , on as , where and are local lightlike bases of and , and and are local orthonormal bases of and , respectively. For this quasiorthonormal fields of frames, we have the following.
Theorem 2.1 (see [8]). Let be an -lightlike submanifold of a semi-Riemannian manifold . Then there exist a complementary vector bundle of 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 the decomposition (2.5), the Gauss and Weingarten formulas are given by where and belong to and , respectively. Here is a torsion-free linear connection on , is a symmetric bilinear form on which is called second fundamental form, and is a linear a operator on and known as shape operator.
According to (2.4) considering the projection morphisms and of on and , respectively, then (2.7) and (2.8) become where we put , and .
As and are -valued and -valued, respectively, therefore they are called the lightlike second fundamental form and the screen second fundamental form on . In particular where , , and . Using (2.9)–(2.12) we obtain for any , , and .
Let be the projection morphism of on ; then using (2.3), 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. and are -valued and -valued bilinear forms and are called as second fundamental forms of distributions and , respectively.
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 . From [8, page 171], using the properties of linear connection, we have Barros and Romero [13] defined indefinite Kaehler manifolds as follows.
Definition 2.2. Let be an indefinite almost Hermitian manifold and let be the Levi-Civita connection on with respect to . Then is called an indefinite Kaehler manifold if is parallel with respect to , that is,
3. Generalized Cauchy-Riemann Lightlike Submanifolds
Definition 3.1. Let be a real lightlike submanifold of an indefinite Kaehler manifold , then is called a generalized Cauchy-Riemann -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 a nondegenerate distribution on , and and are vector bundle of and , respectively.
Then the tangent bundle of is decomposed as is called a proper -lightlike submanifold if , , , and .
Let , , and be the projections on , and , respectively. Then for any we have applying to (3.4), we obtain and we can write (3.5) as where and are the tangential and transversal components of , respectively.
Similarly for any , where and are the sections of and , respectively.
Differentiating (3.5) and using (2.9)–(2.12) and (3.7) we have Using Kaehlerian property of with (2.11) and (2.12), we have the following lemmas.
Lemma 3.2. Let be a -lightlike submanifold of an indefinite Kaehlerian manifold . Then one has where and
Lemma 3.3. Let be a -lightlike submanifold of an indefinite Kaehlerian manifold . Then one has where , , and
Duggal and Sahin [12] investigated the conditions to define totally geodesic foliations by the distributions and in as follows.
Theorem 3.4 (see [12]). Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then the distribution defines a totally geodesic foliation in if and only if , for any .
Theorem 3.5 (see [12]). Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then the distribution defines a totally geodesic foliation in if and only if , for any .
4. Geodesic -Lightlike Submanifolds
Definition 4.1. A -lightlike submanifold of an indefinite Kaehler 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 Kaehler manifold is called geodesic -lightlike submanifold if its second fundamental form satisfies for any .
Definition 4.3. A -lightlike submanifold of an indefinite Kaehler manifold is called geodesic -lightlike submanifold if its second fundamental form satisfies for any .
Theorem 4.4. Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then is -geodesic if and only if for any , and .
Proof. Using the definition of -lightlike submanifolds, is -geodesic, if and only if
for any , and . Thus for , first part of the assertion follows from (2.13).
Now for using (2.16), we have
Since , this implies that , , , or . If or , then we have
and if or , then we have
for any .
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 be a -lightlike submanifold of an indefinite Kaehler manifold . Then is -geodesic if and only if and have no components in for any , , and .
Proof. For and using (2.13), we obtain and for using (2.14) and (2.17) we obtain Hence the assertion follows from (4.7) and (4.8).
Theorem 4.6. Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then is mixed geodesic if and only if for any , and .
Proof. For any , and using (2.14) and (2.17) we obtain and for with (2.13), we obtain Hence the result follows from (4.10) and (4.11).
Theorem 4.7. Let be a mixed geodesic -lightlike submanifold of an indefinite Kaehler manifold . Then one has for any and .
Proof. For and we have Since is mixed geodesic, therefore Using (2.16) and (2.17) we obtain Equating the transversal components we have Thus Now, for and we have If , then using the nondegeneracy of for any , we have . Therefore . Hence the assertion is proved.
Theorem 4.8. Let be a mixed geodesic -lightlike submanifold of an indefinite Kaehler manifold . Then the transversal section is -parallel if and only if , for any .
Proof. Let such that and ; then using hypothesis in (3.9) we have . Now . Since is a Kaehlerian connection and is mixed geodesic, therefore we have or consequently , which clearly proves the theorem.
Theorem 4.9. Let be a -lightlike submanifold of an indefinite Kaehler manifold such that . Then for any and .
Proof. For , and , we have For , and , we have For , and , we have For , and , we have Hence the assertion follows from (4.19)–(4.22).
5. -Lightlike Product
Definition 5.1. A -lightlike submanifold of an indefinite Kaehler manifold is called a -lightlike product if both the distributions and define totally geodesic foliations in .
Lemma 5.2. Let be a totally umbilical -lightlike submanifold of an indefinite Kaehler manifold ; then the distribution defines a totally geodesic foliation in .
Proof. For any , (3.9) implies that ; then for we have where . Since and , then from (3.8) we have , therefore , and this implies that . Therefore (5.1) implies that ; then the nondegeneracy of implies that . Hence , for any . Thus the result follows.
Theorem 5.3. Let be a totally umbilical -lightlike submanifold of an indefinite Kaehler manifold . Then is a -lightlike product if and only if , for any and .
Proof. Let be a -lightlike product; therefore the distributions and define a totally geodesic foliation in . Therefore using Theorem 3.4, for any . Now let and ; then . Hence , for any and .
Conversely, let ; then implies that defines a totally geodesic foliation in . Let ; then (3.9) and (3.11) imply that . Using Lemma 5.2, we obtain , we compare the tangential components, we get , and this implies that . Hence using Theorem 3.5, the distribution defines a totally geodesic foliation in . Consequently, is a -lightlike product of an indefinite Kaehler manifold.
Theorem 5.4. Let be a totally geodesic -lightlike submanifold of an indefinite Kaehler manifold . Suppose that there exists a transversal vector bundle of , which is parallel along with respect to the Levi-Civita connection on ; that is, one has , for any and . Then M is a -lightlike product.
Proof. Since is a totally geodesic -lightlike submanifold, therefore , for any . Hence the distribution defines a totally geodesic foliation in . Next, since for any and , therefore using (2.8), we have then using (3.9), we obtain for any and this implies that . Hence the distribution defines a totally geodesic foliation in . Thus 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 .
Theorem 5.6. Let be an irrotational lightlike submanifold of an indefinite Kaehler manifold . Then is a -lightlike product if the following conditions are satisfied: , .
Proof. Using (2.11) and (2.12) with , we get , and for any and . Therefore using (2.13) we have ; then nondegeneracy of implies that . Hence . Now, let and ; then using , we have . Then using (2.6), we get . Hence . Thus the distribution defines a totally geodesic foliation in .
Next, let ; then . Using (3.9) we obtain , comparing the components along we get , and this implies that . Thus the distribution defines a totally geodesic foliation in . Hence is a -lightlike product.
Theorem 5.7. Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then is a -lightlike product if and only if , for any or .
Proof. Let , for any or . Let , then and (3.9) gives that . Hence using Theorem 3.4, the distribution defines a totally geodesic foliation in . Next, let . Since for any , then (3.9) implies that . Hence using Theorem 3.5, the distribution defines a totally geodesic foliation in . Since both the distributions and define totally geodesic foliations in , hence is a -lightlike product.
Conversely, let be a -lightlike product; therefore the distribution defines a totally geodesic foliation in . Using Kaehlerian property of , for any we have ; then comparing transversal components, we obtain and then , that is, , for any . Let defines a totally geodesic foliation in , and using Kaehlerian property of , we have ; then comparing tangential components on both sides, we obtain ; then (3.9) implies that , which completes the proof.