Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds

In this paper, we introduce and study a new class of CR-lightlike submanifold of an indefinite nearly Sasakian manifold, called quasi generalized Cauchy–Riemann (QGCR) lightlike submanifold. We give some characterization theorems for the existence of QGCR-lightlike submanifolds and finally derive necessary and sufficient conditions for some distributions to be integrable.

the subject, focusing on ascreen and generic lightlike submanifolds [11,12]. Thus, the absence of evidence of research in the geometry of lightlike submanifolds of nearly Sasakian manifolds and the fact that ξ belongs to the tangent bundle of the ambient space have motivated us to introduce a new class of CR-lightlike submanifold of a nearly Sasakian manifold, known as quasi generalized Cauchy-Riemann (QGCR) lightlike submanifold.
The objective of this paper is to characterize totally umbilical and totally geodesic QGCR-lightlike submanifolds of a nearly Sasakian manifold. The rest of the paper is organized as follows. In Sect. 2, we present the basic notions of nearly Sasakian structures and lightlike submanifolds which we refer to in the remaining sections. In Sect. 3, we introduce QGCR-lightlike submanifolds. Section 4 is devoted to the non-existence theorems regarding totally umbilical and totally geodesic QGCR-lightlike submanifolds. Finally, in Sect. 5 we derive the necessary and sufficient conditions for the integrability of the key distributions of a QGCR-lightlike submanifold of an indefinite nearly Sasakian manifold.

Preliminaries
Let M be a (2n + 1)-dimensional manifold endowed with an almost contact structure (φ, ξ, η), i.e., φ is a tensor field of type (1, 1), ξ is a vector field, and η is a 1-form satisfying Then, (φ, ξ, η, g) is called an indefinite almost contact metric structure on M if (φ, ξ, η) is an almost contact structure on M and g is a semi-Riemannian metric on M such that [4] for any vector field X , Y on M, It follows that, for any vector X on M, η(X ) = g(ξ, X ).
If, moreover, for any vector fields X , Y on M, where ∇ is the Levi-Civita connection for the semi-Riemannian metric g, we call M an indefinite nearly Sasakian manifold. We denote by ( ) the set of smooth sections of the vector bundle . Let be the fundamental 2-form of M defined by (X , Y ) = g(X , φ Y ), X , Y ∈ (T M). (2.5) Replacing Y by ξ in (2.4), we obtain for any X ∈ (T M). Introduce a (1,1)-tensor H on M taking for any X ∈ (T M), such that (2.6) reduces to Proof The proof follows from a straightforward calculation.

any X, Y ∈ (T M). Moreover, M is Sasakian if and only if H vanishes identically on M.
Proof The relation (2.10) follows from a straightforward calculation. The second assertion follows from Theorem 3.2 in [2].
Note that, for any X , Y , Z ∈ (T M), This means that the tensor ∇ φ is skew-symmetric. Let (M, g) be an (m + n)-dimensional semi-Riemannian manifold of constant index ν, 1 ≤ ν ≤ m + n and M be a submanifold of M of codimension n. We assume that both m and n are ≥ 1. At a point p ∈ M, we define the orthogonal complement T p M ⊥ of the tangent space T p M by defines a smooth distribution on M of rank r > 0. We call Rad T M the radical distribution on M. In the sequel, an r -lightlike submanifold will simply be called a lightlike submanifold and g is lightlike metric, unless we need to specify r . (2.14) Note that the distribution S(T M) is not unique and is canonically isomorphic to the factor vector bundle T M/Rad T M [10]. We say that a lightlike submanifold M of M is Similarly to [11], we use the following range of indices in this paper, where {W 1+r , . . . , W n } is an othornomal basis of (S(T M ⊥ )| U ) and let α = g(W α , W α ) be the signatures of W α .
Let P be the projection morphism of T M on to S(T M). Using the decomposition (2.14), consider the projection morphisms L and S of tr(T M) on ltr(T M) and S(T M ⊥ ), respectively. Then, the Gauss-Wiengartein equations [9] of an r -lightlike submanifold M and S(T M) are the following:  [7,9] that for any X, Y ∈ (T M). It is easy to see that ∇ * is a metric connection on S(T M), while ∇ is generally not a metric connection and is given by for any X, Y ∈ (T M) and λ i are 1-forms given by The above three local second fundamental forms are related to their shape operators by the following set of equations Moreover, it is easy to see that M is totally umbilical in M, if and only if on each coordinate neighborhood U there exist smooth vector fields H l ∈ (ltr(T M)) and H s ∈ (S(T M ⊥ )) and smooth functions H l i ∈ F(ltr(T M)) and H s α ∈ F(S(T M ⊥ )), such that The above definition is independent of the choice of the screen distribution.

Quasi generalized CR-lightlike submanifolds
Generally, the structure vector field ξ belongs to T M. Therefore, we define it according to decomposition (2.14) as follows; where ξ S is a smooth vector field of S(T M) and ξ S ⊥ , ξ R , ξ l are defined as follows Generalized Cauchy Riemann (GCR) lightlike submanifolds were introduced in [9,10], in which the structure vector field ξ was assumed tangent to the submanifold. Contrary to this assumption, we introduce a special class of C R-lightlike submanifold in which ξ belongs to T M, called quasi generalized Cauchy-Riemann (QGCR)-lightlike submanifold as follows.
where D 0 is a non-degenerate distribution on M and L and S are respectively vector subbundles of ltr(T M) and S(T M ⊥ ).
Let M be a QGCR-lightlike submanifold of an indefinite nearly Sasakian manifold M. If the structure vector field ξ is tangent to M, then ξ ∈ (S(T M)). The proof of this is similar to the one given by Calin in the Sasakian case [5]. In this case, if X ∈ (S) and Y ∈ (L), then η(X ) = η(Y ) = 0 and where ξ is the 1-dimensional distribution spanned by ξ . Therefore, the QGCR-lightlike submanifold tangent to ξ reverts to a GCR-lightlike submanifold [10].

Proposition 3.2 A QGCR-lightlike submanifold M of an indefinite nearly Sasakian manifold M tangent to the structure vector field ξ is a GCR-lightlike submanifold.
Next, we follow Yano and Kon [18, p. 353] definition of contact CR-submanifolds and state the following definition for a quasi contact CR-lightlike submanifold.

Definition 3.3 Let (M, g, S(T M), S(T M ⊥ )
) be a lightlike submanifold of an indefinite nearly Sasakian manifold (M, g, φ, ξ, η). We say that M is quasi contact CR-lightlike submanifold of M if the following conditions are satisfied: where D 0 is non-degenerate; L 1 is a vector subbundle of S(T M ⊥ ).
It is easy to see that when the structure vector field ξ is tangent to the quasi contact CR-lightlike submanifold M, then M is a contact CR.

Proposition 3.4 A QGCR-lightlike submanifold of an indefinite nearly Sasakian manifold M is a quasi contact
Notice that D is invariant with respect to φ, while D is not generally anti-invariant with respect to φ. Note the following for a proper QGCR-lightlike submanifold (M, g, S(T M), S(T M ⊥ )) of an indefinite almost contact metric manifolds M according to Definition 3.1: Next, we adopt the definition of ascreen lightlike submanifolds used by Jin [12] for the case of contact ambient manifold in case of lightlike submanifolds of an almost contact manifold. It is easy to check that the complementary subbundle ν is invariant under φ, i.e., φν = ν.
Let M be an ascreen QGCR-lightlike submanifold of an indefinite nearly Sasakian manifold M. Then by Definition 3.5, the structural vector field ξ ∈ Rad T M ⊕ltr(T M). This means that ξ is either in Rad T M or ltr(T M). If ξ ∈ Rad T M, then ξ ∈ D 2 since φ D 1 = D 1 and φξ = 0. On the other hand, if ξ ∈ ltr(T M), then ξ ∈ (L) because of the fact that φν = ν and φξ = 0. Therefore, we have the following.  where E ∈ (D 2 ) and N ∈ (L), and a = η(N ) and b = η(E) are non-zero smooth functions. Applying φ to the first relation of (3.8) and using the fact that φξ = 0, we get Conversely, suppose that φL = φ D 2 . Then, there exists a non-vanishing smooth function ω such that Taking the g-product of (3.10) with respect to φ E and φ N in turn, we get b 2 = ω(ab − 1) and ωa 2 = ab − 1. (3.11) Since ω = 0, by (3.11), we have a = 0, b = 0 and b 2 = (ωa) 2 . The latter gives b = ±ωa. The case b = ωa implies that ab = ωa 2 = ab − 1, which is a contradiction. Thus b = −ωa, from which 2ab = 1. Since ω = − b a , a = 0 and φ E = ωφ N , it is easy to see that aφ E + bφ N = 0. Applying φ to this equation, and using the first relation in (2.1), together with 2ab = 1, we get ξ = a E + bN . Therefore, M is ascreen lightlike submanifold of M.
We notice from the above theorem that if ξ is tangent to M, thenḡ(H φW α , W α ) = 0. It is easy to see that g(H X, W α ) = 0, for all X ∈ (φS). Hence, H X has no component along S for all X ∈ (φS). When the structure vector field ξ is normal, we have the following. Differentiating the first equation of (4.7) with respect to X and using (2.6), (2.15) and (2.18), we get for all X ∈ (T M). Taking the g-product of (4.8) with respect to E k and φ N k ∈ (S(T M)) in turn, where N k ∈ (L), we get Replacing X with φ N k in (4.9), we obtain (4.10) The g-product with φ N k yields Now, using (2.22)-(2.24) in (4.11), we obtain which on replacing X with E k and simplifying gives Adding (4.10) to (4.12) yields But H is skew-symmetric and thus (4.13) becomes (4.14) By virtue of (4.14), it is easy to see that H φ N k ∈ (ltr(T M)), particularly, in the direction of N k . Hence, there exists a non-vanishing smooth function f k such that H φ N k = f k N k . Taking the g-product of this equation with respect to ξ , we get 0 = g(H φ N k , ξ) = g( f k N k , ξ) = f k g(N k , ξ) = f k a k , from which a k = 0, a contradiction. Therefore, in a proper QGCR-lightlike submanifolds of an indefinite nearly Sasakian manifold, ξ does not belong to T M ⊥ .
In particular, we have the following. (1) H X belongs to ltr(T M) for all X ∈ (φL).
(2) H X belongs to Rad T M for all X ∈ (φ D 2 ). Differentiating the first equation of (4.15) with respect to X and using (2.6), (2.16) and (2.18), we get for all X ∈ (T M). Now, taking the g-product of the above equation with respect to φ N k ∈ (S(T M)) where N k ∈ (L), we get Substituting (2.23) and the first equation of (2.26) in (4.17) give Since M is totally umbilical in M, with a totally umbilical screen, (4.18) yields which reduces to g(E k , N k ) = g(φ H E k , N k ) = 1. It is easy to see from this equation that φ H E k ∈ (Rad T M). In particular, there exist non-vanishing smooth functions w k such that φ H E k = w k E k . Taking the g-product of this last equation with respect to ξ , we obtain 0 = g(φ H E k , ξ) = w k g(E k , ξ) = w k b k . Hence, b k = 0, and this contradiction completes the proof. Next, we consider the special case H = 0. In particular, the indefinite nearly Sasakian manifold (M, φ, η, ξ, g) with H = 0 becomes Sasakian. An indefinite Sasakian manifold M is called an indefinite Sasakian space form, denoted by M(c), if it has a constant φ-sectional curvature c [13]. The curvature tensor R of the indefinite space form M(c) is given by for any X , Y , Z ∈ (T M). Now, using (4.20) we have the following existence theorem.
Since g(φ X, Y ) = 0 and g(φ E, N ) = 0, we have g(R(X, Y )E, N ) = 0. Similarly, from (4.20), one obtains Note that conditions (b) and (c) are independent of the position of ξ and hence valid for GCR-lightlike submanifolds [10] and QGCR-lightlike submanifolds of an indefinte Sasakian space form M(c). When ξ is tangent to M, it is well known [5] that ξ ∈ (S(T M)). In this case, one has a GCR-lightlike submanifold, in which D 2 ⊥ φ D 2 is an invariant subbundle of T M, leading to D = D 1 ⊥ D 2 ⊥ φ D 2 ⊥ D 0 as the maximal invariant subspace of T M. On the other hand, when M is QGCR-lightlike submanifold, then ξ ∈ (T M) and thus D 2 ⊥ φ D 2 is generally not an invariant subbundle of T M, since the action of φ on it gives a component along ξ . In particular, let E ∈ (D 2 ) then E + φ E ∈ (D 2 ⊥ φ D 2 ). But on applying φ to this subbundle and considering the fact that η(E) = 0, we get

Integrability of the distributions D and D
Let M be a QGCR-lightlike submanifold of an indefinite nearly Sasakian manifold (M, g, φ, ξ, η). From (2.12), the tangent bundle of any QGCR lightlike submanifold, T M, can be rewritten as Notice that D is invariant with respect to φ, while D is not generally anti-invariant with respect to φ. Let π and π be the projections of T M onto D and D, respectively. Then, using the first equation of (5.1), we can decompose X as X = π X + π X, ∀X ∈ (T M).
It is easy to see that φπ X ∈ (D). However, the action of φ on π X gives a tangential and transversal component due to a generalized ξ , i.e., φ X = P 1 X + P 2 X + Q X, ∀X ∈ (T M), where P 1 X = φπ X while P 2 X is the tangential component of φ π X and Q X is the transversal component of φ X , essentially coming from φ π X since φ D = D. By grouping the tangential and transversal parts in (5.3), it is easy to see that where P X = P 1 X + P 2 X . Note that if X ∈ (D), then P 2 X = Q X = 0, and φ X = P 1 X . The Eq. (5.4) can be properly understood through the following specific case of vector field in D ⊂ D. Let ξ M and ξ trM be the tangential and transversal components of ξ . If X ∈ (D) and since D = φ S ⊕ φ L, then Consequently, for X ∈ (D), Similarly, for any V ∈ (tr(T M)), V = SV + LV , and where t V and f V are the tangential and transversal components of φV , respectively. Differentiating (5.4) with respect to Y, we get and from (2.4), we have Finally putting (5.7) and (5.8) in (5.6) and then comparing the tangential and transversal components of the resulting equation, we obtain (∇ Y P)X + (∇ X P)Y = A Q X Y + A QY X + 2th(X, Y ) + 2g(X, Y )ξ M − η(X )Y − η(Y )X, (5.9) and for all X, Y ∈ (T M).
Proof The proof follows from (5.9) and (5.10). Proof The proof is a straightforward calculation.
The integrability of D is discussed as follows. Note that the distribution D is integrable if and only if, for any X , Y ∈ ( D), [X, Y ] ∈ ( D). The latter is equivalent to P 1 [X, Y ] = 0.
Proof Let X, Y ∈ ( D), then it is easy to see that P 1 X = P 1 Y = 0. Hence, P X = P 2 X and PY = P 2 Y . Now using (5.12), we derive 14) It is obvious from (5.14) that the last four terms belong to D. Hence, the assertation follows from the remaining terms.