Abstract

Attention is drawn to some distributions on ascreen Quasi-Generalized Cauchy-Riemannian (QGCR) null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifolds.

1. Introduction

One of the current interesting research areas in semi-Riemannian geometry is the theory of null (or lightlike) submanifolds. An intrinsic approach to the theory of null submanifolds was advanced by Kupeli [1], yet an extrinsic counterpart had to wait for Duggal and Bejancu [2], and later by Duggal and Sahin [3]. Since then, many researchers have laboured to extend their theories with evidence from the following few selected papers: [3–11] and other references therein. The rapid increase in research on this topic, since 1996, is inspired by the numerous applications of the theory to mathematical physics, particularly in general relativity. More precisely, in general relativity, null submanifolds represent different models of black hole horizons (see [2, 3] for details).

In [5], the authors initiated the study of generalized CR- (GCR-) null submanifolds of an indefinite Sasakian manifold, which are tangent to the structure vector field, , of the almost contact structure . Moreover, when is tangent to the submanifold, Calin [12] proved that it belongs to its screen distribution. This assumption is widely accepted and it has been applied in many papers on null contact geometry, for instance, [3, 5, 8–11, 13]. It is worth mentioning that is a global vector field defined on the entire tangent bundle of the ambient almost contact manifold. Thus, restricting it to the screen distribution is only one of those cases in which it can be placed. In the study of Riemannian CR-submanifolds of Sasakian manifolds, Yano and Kon [14, p. 43] proved that making a normal vector field in such scenario leads to an anti-invariant submanifold, and hence was kept tangent to the CR-submanifold. Their proof leans against the fact that the shape operator on such CR-submanifold is naturally symmetric with respect to the induced Riemannian metric . On the other hand, the shape operators of any -null submanifold are generally not symmetric with respect to the induced degenerate metric (see [2, 3] for details).

In an attempt to generalize , we introduced a special class of CR-null submanifolds of a nearly Sasakian manifold, known as quasi-generalized CR- (QGCR-) null submanifold [15], for which the classical GCR-null submanifolds [3] form part. Among other benefits, generalizing leads to QGCR-null submanifolds of lower dimensions and with quite different geometric properties compared to respective GCR-null submanifolds.

The purpose of this paper is to investigate the geometry of distributions on ascreen QGCR-null submanifolds of indefinite nearly cosymplectic manifolds. A null submanifold of an indefinite nearly cosymplectic manifold is called ascreen if the structure vector field belongs to [16]. The paper is organized as follows. In Section 2, we present the basic notions of null submanifolds and nearly cosymplectic manifolds. More details can be found in [17–22]. In Section 3, we review the basic notions of QGCR-null submanifolds and we give an example of ascreen QGCR-null submanifold. In Section 4, we discuss totally umbilical, totally geodesic, and irrotational ascreen QGCR-null submanifolds of an indefinite nearly cosymplectic space form . Finally, in Section 5 we investigate the geodesity of the distributions and .

2. Preliminaries

Let be a codimension submanifold of a semi-Riemannian manifold of constant index , , with . Then, is said to be a null submanifold of if the tangent and normal bundles of have a nontrivial intersection. This intersection defines a smooth distribution on , called the radical distribution [2]. More precisely, consider ; one defines the orthogonal complement of the tangent space by If we denote the radical distribution on by , then . The submanifold of is said to be -null submanifold (one supposes that the index of is ), if the mapping defines a smooth distribution on of rank .

In this paper, -null submanifold will simply be called a null submanifold and is a null metric, unless we need to specify .

Let be a screen distribution which is a semi-Riemannian complementary distribution of in ; that is,Consider a screen transversal bundle , which is semi-Riemannian and complementary to in . For any local basis of , there exists a local null frame in such that and . It follows that there exists a null transversal vector bundle locally spanned by (see details in [2, 3]). If denotes the complementary (but not orthogonal) vector bundle to in , then,It is important to note that the screen distribution is not unique and is canonically isomorphic to the factor vector bundle [1].

Given a null submanifold , then the following classifications of are well-known [2]: (i) is -null if ; (ii) is coisotropic if , ; (iii) is isotropic if , ; (iv) is totally null if , .

Where necessary, the following range of indices will be used Consider a local quasi-orthonormal fields of frames of along as where and are, respectively, orthonormal bases of and .

Throughout the paper we consider to be a set of smooth sections of the vector bundle .

Let be the projection morphism of onto . Then, the following are the Gauss-Weingarten equations of -null submanifold and (see [2, 3] for detailed explanations):where and are the induced connections on and , respectively and and are symmetric bilinear forms known as local null and screen fundamental forms of , respectively. Furthermore, are the second fundamental forms of . , , and are linear operators on while , , , and are 1-forms on . Note that the second fundamental tensor of is given byfor any . The connection is a metric connection on while is generally not a metric connection and is given by for any and are 1-forms given by , for all By using (7), (8), and (9), the curvature tensors and of and , respectively, are related as, for any ,A null submanifold of an indefinite manifold is said to be totally umbilical in [3] if there is a smooth transversal vector field , called the transversal curvature vector of such thatMoreover, it is easy to see that is totally umbilical in , if and only if on each coordinate neighborhood there exist smooth vector fields and and smooth functions and such that, for all .

Let us now consider to be a -dimensional manifold endowed with an almost contact structure ; that is, is a tensor field of type , is a vector field, and is a 1-form satisfyingThen is called an indefinite almost contact metric structure on if is an almost contact structure on and is a semi-Riemannian metric on such that [19], for any vector field , on ,An indefinite almost contact metric manifold is said to be nearly cosymplectic ifwhere is the Levi-Civita connection for . Taking in (19), we getIt is easy to verify the following properties of : for all . Let denote the fundamental 2-form of defined by Then the 1-form and tensor are related as follows.

Lemma 1. Let be indefinite nearly cosymplectic. Then,Moreover, is cosymplectic if and only if vanishes identically on .

Note that, for all , , , which means that the tensor is skew-symmetric. The following lemma is fundamental to the sequel.

Lemma 2. Let be a nearly cosymplectic manifold. Thenfor all .

Proof. The proof follows from a straightforward calculation.

3. Quasi-Generalized CR-Null Submanifolds

We recall some basic notions on QGCR-null submanifolds (see [15] for details).

The structure vector field of an indefinite almost contact manifold can be written according to decomposition (4) as follows:where is a smooth vector field of while , , and , all smooth functions on . Here .

We adopt the definition of quasi-generalized CR- (QGCR-) null submanifolds given in [15] for indefinite nearly cosymplectic manifolds.

Definition 3. Let be a null submanifold of an indefinite nearly cosymplectic manifold . We say that is quasi-generalized CR- (QGCR-) null submanifold of if the following conditions are satisfied.
(i) There exist two distributions and of such that(ii) There exist vector bundles and over such thatwhere is a nondegenerate distribution on and and are, respectively, vector subbundles of and .

If , , , and , then is called a proper QGCR-null submanifold.

A proof of the following proposition uses similar arguments as in [15].

Proposition 4. A QGCR-null submanifold of an indefinite nearly cosymplectic manifold tangent to the structure vector field is a GCR-null submanifold.

Using (2), the tangent bundle of any QGCR-null submanifold, , can be decomposed as Unlike a GCR-null submanifold, in a QGCR-null submanifold, is invariant with respect to while is not generally anti-invariant.

Throughout this paper, we suppose that is a proper QGCR-null submanifold. From the above definition, we can easily deduce the following:(1)Condition (i) implies that .(2)Condition (ii) implies that and .

Definition 5 (see [16]). A null submanifold of a semi-Riemannian manifold is said to be ascreen if the structure vector field, , belongs to .

From Definition , Lemma , and Theorem of [15], we have the following.

Theorem 6. Let be an ascreen QGCR-null submanifold of an indefinite nearly cosymplectic manifold ; then . If is a 3-null QGCR submanifold of an indefinite nearly cosymplectic manifold , then is ascreen null submanifold if and only if .

Proof. The proof follows from straightforward calculation as in [15].

It is crucial to note the following aspects with ascreen QGCR-null submanifold: item (2) of Definition 3 implies that and . Thus and , and any 7-dimensional ascreen QGCR-null submanifold is 3-null.

In what follows, we construct an ascreen QGCR-null submanifold of a special nearly cosymplectic manifold with (i.e., is a cosymplectic manifold). Thus, let denote the manifold with its usual cosymplectic structure given by where are Cartesian coordinates and , for .

Now, we use the above structure to construct the following example.

Example 7. Let be a semi-Euclidean space, with being of signature with respect to the canonical basis Let be a submanifold of given by where . By direct calculations, we can see that the vector fields form a local frame of . Then is spanned by , and, therefore, is 3-null. Further, ; therefore we set . Also and thus . It is easy to see that , so we set . On the other hand, following direct calculations, we have from which and . Clearly, . Further, and thus . Notice that and therefore . Also, and therefore . Finally, we calculate as follows. Using Theorem 6 we have . Applying to this equation we obtain . Now, substituting for and in this equation we get , from which we get . Since and , we conclude that is an ascreen QGCR-null submanifold of .

Proposition 8. There exist no coisotropic, isotropic, or totally null proper QGCR-null submanifolds of an indefinite nearly cosymplectic manifold.

4. Umbilical and Geodesic Ascreen QGCR-Null Submanifolds

In this section, we prove two main theorems concerning totally umbilical, totally geodesic and irrotational ascreen QGCR-null submanifolds of . An indefinite nearly cosymplectic manifold is called an indefinite nearly cosymplectic space form, denoted by , if it has the constant -sectional curvature . The curvature tensor of the indefinite nearly cosymplectic space form is given by [21]:for all .