Bejancu [1] initiated the study of CR-submanifolds with Kaehler manifolds. Yano and Kon [10] introduced the concept of odd dimensional manifolds called the contact CR submanifolds of a Sasakian manifold. They proved some basic results for contact CR submanifolds of a Sasakian manifold with definite metric. Further, Matsumoto [8] studied such manifolds and obtained some fundamental results. Ganchev et al. [6] introduced the geometry of almost contact B metric manifolds which are a natural extension, to the odd dimensional case, of the geometry of the almost complex manifolds with B-metric.

Duggal and Sahin [5] defined and studied lightlike submanifolds of indefinite Sasakian manifolds and introduced radical tranversal lightlike submanifolds of indefinite Sasakian manifolds. Recently, Nakova [9] studied submanifolds of almost complex manifolds with Norden metric which are non-degenerate with respect to one Norden metric and lightlike with respect to other Norden metric on the manifold and introduced radical transversal lightlike submanifold of almost complex manifolds with the Norden metric. Ivanov et al[7] defined Sasaki-like almost contact Complex Riemannian manifolds that resemble with the Sasakian manifold and thus motivated us to study such manifolds.

In this paper, we study the geometry of contact CR submanifolds and radical transversal lightlike submanifolds of Sasaki-like almost contact manifold with B-metric. We investigate conditions for the integrability of distributions of contact CR-submanifolds and radical transversal lightlike submanifolds of Sasaki like almost contact manifold with B-metric. We find the necessary and sufficient condition for integrability of screen distribution of radical transversal lightlike submanifold with B-metric. Further, we obtain some new results that establish relationship between the concerned geometric objects of both the submanifolds of Sasaki-like almost contact manifold with B-metric. Finally, we prove that for a contact CR-product submanifold of Sasaki like almost contact manifold with B-metric, the induced connection ∇̃ of radical transversal lightlike submanifold with Norden metric is a metric connection.

Let (M̄²ⁿ⁺¹, ϕ, ζ, η) be an almost contact manifold with B-metric [6], that is, let (ϕ, ζ, η) be an almost contact structure consisting of a tensor field ϕ of type (1, 1) a vector field ζ, a 1-form η and a metric ḡ on M̄ satisfying the following algebraic conditions for arbitrary vector fields X and Y on M̄:

The following identities are valid for an almost contact manifold with B-metric.

The associated metric g˜¯ of ḡ on M̄ defined by

is also a B-metric on M̄ and the manifold (M̄, ϕ, ζ, η, g˜¯) is also called an almost contact manifold with B-metric. Both the metrics ḡ and g˜¯ are indefinite of signature (n + 1, n).

Let ∇¯and ∇˜¯ be the Levi-Civita connection of ḡ and g˜¯ respectively. The tensor field F of type (0, 3) is defined on M̄ by

and the following general properties hold [6]:

for any X, Y, Z ∈ T M̄. The relations of F with ∇ζ and ∇η are given by :

Let {e_i, ζ}, (i = 1, 2, ....., 2n) be a basis of T_mM and (g^ij) be the inverse matrix of g_ij then for X ∈ T_mM̄, the following 1-forms are associated with F :

Using above equation, we have

The Nijenhuis tensor N of the almost complex structure is defined by

An almost contact structure (ϕ, ζ, η) is said to be normal if and only if Nijenhuis tensor denoted by N vanishes [2] and such manifold (M̄,ϕ, ζ, η, ḡ;) is called normal almost contact manifold.

In [6] Ganchev et al. defined eleven basic classes F_i(i = 1, 2, ....11) of almost contact manifolds with B-metric and gave a classification of almost contact manifolds with B-metric with respect to tensor F. The special class F₀ defined by the condition F(X, Y, Z) = 0 belongs to everyone of the basic classes. Throughout this paper, we will consider the class F₀.

Definition 2.1.([7])

An almost contact manifold (M̄,ϕ, ζ, η, ḡ) with B- metric is Sasaki-like if the structure tensors ϕ, ζ, η, ḡ satisfy the following equalities

Also, the covariant derivative ∇¯ϕ satisfies the following equality

In this paper, we refer to these manifolds as indefinite Sasaki-like almost contact manifolds with B-metric.

In this section, we define and study contact CR-submanifolds of Sasaki-like almost contact manifolds with B-metric and investigate their integrability conditions.

Definition 3.1

A submanifold (M, g) of a (2n+1)-dimensional Sasaki-like almost contact manifold with B-metric M̄ with structure tensors (ϕ, ζ, η) is called a contact CR-submanifold if there exists a differentiable distribution D : x → D_x ⫅ T_xM and the complementary orthogonal distribution D⊥:x→Dx⊥⊂TxM on M which satisfies the following conditions:

• (i) ζ ∈ D,

• (ii) ϕD_x ⊂ T_xM for each x ∈ M,

• (iii) ϕDx⊥⫅TxM⊥ for each point x ∈ M.

The complementary orthogonal distribution of ϕD^⊥ in TM^⊥ is denoted by π. The tangent bundle T M̄ of M̄ has the following decomposition

Let E and G be the projection morphisms of TM on the distributions D and D^⊥ respectively, then for any X ∈ TM, we can write

where EX ∈ D and GX ∈ D^⊥. Applying ϕ to (3.1), we get

where ϕEX = HX ∈ D and ϕGX = KX ∈ ϕD^⊥ are the tangential and the normal components of ϕX, respectively.

Similarly for any V ∈ TM^⊥, we have

where tV and fV are the tangential and the normal parts of ϕV, respectively.

Let ∇¯ and ∇ be the Levi-Civita connections of ḡ and g on M̄ and M, respectively. Then the Gauss and Weingarten formulae for M̄ are given by

for any vector fields X, Y ∈ TM and V ∈ TM^⊥, where h is the second fundamental form of M, A_V is the shape operator of M with respect to V and D is the normal connection on TM^⊥ which is a metric linear connection. Since ζ ∈ TM, we have, for any vector field X ∈ TM,

and on equating the components of ϕX, we get

Let E₁ and E₂ be the projection morphisms of TM^⊥ on ϕD^⊥ and π, respectively. Then (3.2) and (3.3) can be written as

where

It should be noted that D¹ and D² are not linear connections on TM^⊥ but are Otsuki connections with respect to the vector bundle morphisms E₁ and E₂ respectively. Thus the above equation (3.6) reduces to

where ∇X1N=DX1N and ∇X2W=DX2W are metric connections on ϕD^⊥ and π, respectively and D1(X,W)=DX1W and D2(X,N)=DX2N are F(M̄) bilinear mappings. Making use of equations (3.7) to (3.9), we have

Lemma 3.1

Let (M, g) be a contact CR-submanifold of an indefinite Sasaki-like almost contact manifold (M̄,ϕ, ζ, η, ḡ) with B-metric. Then we have

where

for any X, Y ∈ TM.

Lemma 3.2

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold (M̄,ϕ, ζ, η, ḡ) with B-metric. Then we have

for any X, Y ∈ TM, N ∈ ϕD^⊥and W ∈ π.

Theorem 3.3

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold (M̄,ϕ, ζ, η, ḡ with B-metric. Then ∇¹K = 0 if and only if ∇¹t = 0.

Lemma 3.4

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold (M̄, ϕ, ζ, η, ḡ) with B-metric. Then we have

and

for any Z, W ∈ D^⊥.

Lemma 3.5

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold (M̄,ϕ, ζ, η, ḡ) with B-metric. Then the anti-invariant distribution D^⊥is integrable.

Lemma 3.6

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold (M̄,ϕ, ζ, η, ḡ) with B-metric. Then the distribution D is integrable if and only if h¹(X,ϕY ) = h¹(Y,ϕX).

Theorem 3.7

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold (M̄,ϕ, ζ, η, ḡ) with B-metric. Then (M, g) is a contact CR-product if and only if

∀X ∈ D^⊥and Z ∈ D.

Theorem 3.8

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold (M̄,ϕ, ζ, η, ḡ) with B-metric. Then the following assertions are equivalent :

• (i) D¹is a metric Otsuki connection on TM^⊥.

• (ii) D¹(X, W) = 0 for any X ∈ TM and W ∈ π.

• (iii) D²(X, N) = 0 for any X ∈ TM and N ∈ ϕD^⊥.

• (iv) D²is a metric Otsuki connection on TM^⊥.

Consider an m-dimensional submanifold (M, g) immersed in a real (m + n)-dimensional semi-Riemannian manifold (M̄, ḡ) of constant index q such that m, n ≥ 1, 1 ≤ q ≤ m+ n − 1 and let g be the induced metric of ḡ on M. Then M is called a lightlike submanifold of M̄ if ḡ is a degenerate metric on the tangent bundle TM of M. For a degenerate metric g on M, TM^⊥ is a degenerate n-dimensional subspace of T_xM̄. Thus both T_xM and T_xM^⊥ are no longer complementary but degenerate orthogonal subspaces of T M̄. So, there exists a subspace called radical or null subspace, that is,

Further, if the mapping Rad(TM) : x ∈ M → RadT_xM, defines a smooth distribution of rank r > 0 on M then the submanifold M is called an r-lightlike submanifold of M̄ and Rad(TM) is known as the radical distribution on M. A semi-Riemannian complementary distribution S(TM) of Rad(TM) in TM is a screen distribution. We have

and that 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 M̄ |_M and to Rad(TM) in S(TM^⊥)^⊥ respectively. Then, we have

For a quasi-orthonormal fields of frames of M̄ along M, we have the following.

Theorem 4.1.([4])

Let (M, g, S(TM), S(TM^⊥)) be an r-lightlike submanifold of a semi-Riemannian manifold (M̄, ḡ). Then there exists a complementary vector bundle ltr(TM) of Rad(TM) in S(TM^⊥)^⊥and a basis of ltr(TM) |_Uconsisting of smooth section {N_a} of S(TM^⊥)^⊥|_U, whereUis a coordinate neighbourhood of M, such that

where {ξ₁, ..., ξ_r} is a lightlike basis of Rad(TM).

Let (M, g̃) be an r-lightlike submanifold of a Sasaki-like almost contact manifold with B metric(M̄, ϕ, ḡ, g˜¯). Let ∇˜¯ be the Levi-Civita connection of the metric g˜¯ on M̄ and ∇˜be the induced connection on M then Gauss and Weingarten formulae are given by ∇˜¯XY=∇˜XY+h˜(X,Y),         ∇˜¯XV=-A˜VX+∇XtV,

for arbitrary X, Y ∈ TM and V ∈ tr(TM), where {∇̃_XY, Ã_VX} and {h̃(X, Y ),{h˜(X,Y),∇XtV} } belong to TM and tr(TM), respectively and ∇̃; and ∇^t are linear connections on TM and tr(TM), respectively. Moreover, ∇̃ is torsion-free linear connection, h̃ is a tr(TM)-valued symmetric ℱ(M)-bilinear form on TM and à is a TM-valued ℱ(M)-bilinear form on tr(TM) × (TM). In general, ∇˜and ∇^t are not metric connections. Let L and S be the projection morphisms of tr(TM) on ltr(TM) and S(TM^⊥̃), respectively then

where

Besides D^l and D^s do not define linear connections on tr(TM) but they are Otsuki connections on tr(TM) with respect to L and S, respectively. Therefore (4.1) and (4.2) become

where ∇^l and ∇^s are defined by ∇XlN=DXlN and ∇XsW=DXsW are metric linear connections on ltr(TM) and S(TM^⊥̃), respectively. D^l and D^s are defined by Dl(X,W)=DXlW and Ds(X,N)=DXsN are ℱ(M)-bilinear mappings. Using (4.3)– – (4.5) and taking into account that ∇˜¯ is a metric connection, we obtain

Let P′ be the projection morphism of TM on S(TM) then new induced geometric objects on the screen distribution S(TM) are given as below.

for any X, Y ∈ TM and ξ ∈ Rad(TM), where {∇X*P′Y,Aξ*X } and {h^*(X, P′ Y ), ∇X*tξ } belong to S(TM) and Rad(TM), respectively. ∇^* and ∇^*t are linear connections on complementary distributions S(TM) and Rad(TM), respectively. h^* and A^* are Rad(TM)-valued and S(TM)-valued bilinear forms and they are called as the second fundamental forms of distributions S(TM) and Rad(TM), respectively, for any X, Y ∈ TM, ξ ∈ Rad(TM) and N ∈ ltr(TM).

From the geometry of Riemannian submanifolds, it is known that on a non degenerate submanifold the induced connection ∇ is a metric connection. But unfortunately, this is not true incase of a lightlike submanifold. Indeed, considering ∇̄ a metric connection, we have

for any X, Y, Z ∈ TM.

In [11], Sahin defined radical transversal lightlike submanifolds for an indefinite Sasakian manifold. In this paper, we define radical transversal lightlike submanifold of a Sasaki-like almost contact manifold with B-metric as follows.

Definition 4.2

Let (M, g̃, S(TM), S(TM)^⊥) be a lightlike submanifold of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, ḡ, g˜¯). Then M is a radical transversal lightlike submanifold of M̄ if

It is important to note that for an indefinite Sasakian manifold [11] there do not exist any 1-lightlike radical transversal lightlike submanifolds. But for a Sasaki-like almost contact manifold with B-metric there exists an 1-lightlike radical transversal lighlike submanifold.

Lemma 4.3

There exists an 1-lightlike radical transversal lighlike submanifold (M, g̃) of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, ḡ, g˜¯).

Theorem 4.4

Let (M, g̃) be a radical lightlike submanifold of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, η, ζ, ḡ, g˜¯). Then the induced metric connection ∇̃ on M is a metric connection if and only if A_ϕYX has no component in S(TM) for X ∈ TM and Y ∈ Rad(TM).

Since there are two B-metrics on an indefinite almost contact manifold M̄ with B-metric, there are two corresponding induced metrics on the submanifold M of M̄. Hence M is either non-degenerate (degenerate) with respect to both the induced metrics or degenerate with respect to one and non-degenerate with respect to other. In [9], Nakova studied the case when the submanifold (M, g) is non-degenerate and (M, g̃) is a degenerate submanifold of M̄. In particular, Nakova proved the following theorem for an almost complex manifold with a Norden metric.

Theorem 5.1.([9])

Let (M̄, J̄, ḡ, g˜¯) be a 2n-dimensional almost complex manifold with a Norden metric and M be an m-dimensional submanifold of M̄. The submanifold (M, g) is a CR-submanifold with an r-dimensional totally real distribution D^⊥if and only if (M, g̃) is an r-lightlike radical transversal lightlike submanifold of M̄.

Thus, in this case, the tangent bundle T M̄ of M̄ has the following decomposition.

where S(TM) = D, RadTM = D^⊥, S(TM^⊥̃) = (ϕD^⊥)^⊥ and ltrTM = ϕD^⊥. For a Sasaki-like almost contact manifold with B-metric, we have

We have verify the above result for a Sasaki-like almost contact manifold with B-metric and the proof is same as above theorem if we consider that ζ ∈ S(TM) [3].

Theorem 5.2

Let (M, g) be a submanifold of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, η, ζ, ḡ, g˜¯). Then the submanifold (M, g) is a contact CR-submanifold with an r-dimensional totally real distribution if and only if (M, g̃) is an r-lightlike contact radical transversal lightlike submanifold of M̄.

Theorem 5.3

Let (M, g̃) be a radical transversal lightlike submanifold of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, η, ζ, ḡ, g˜¯). Then the radical distribution of (M, g̃) is integrable.

Theorem 5.4

Let (M̄, ϕ, η, ζ, ḡ, g˜¯) be a Sasaki-like almost contact manifold with B-metric and (M, g) be a contact CR submanifold of M̄. Then the invariant distribution D of contact CR-submanifold (M, g) is integrable if and only if the screen distribution S(TM) of radical transversal lightlike submanifold (M, g̃) is integrable.

Theorem 5.5

Let (M, g) be a contact CR-submanifold of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, η, ζ, ḡ, g˜¯). Then the induced connection ∇̃ on the radical transversal lightlike submanifold (M, g̃) is a metric connection if and only if (M, g) is a contact CR-product.

Theorem 5.6

Let (M, g) be a contact CR-submanifold of a of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, η, ζ, ḡ, g˜¯). If K is parallel, that is, ∇¹K = 0 then the screen distribution S(TM) of radical transversal lightlike submanifold (M, g̃) is a parallel distribution with respect to ∇̃ and h^*vanishes identically on lightlike submanifold (M, g̃).

Theorem 5.7

Let (M, g) be a contact CR-submanifold of of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, η, ζ, ḡ, g˜¯). If(∇X1t)N=0then the screen distribution S(TM) of radical transversal lightlike submanifold (M, g̃) is a parallel distribution with respect to ∇̃ and h^*vanishes identically on lightlike submanifold (M, g̃).

Theorem 5.8

Let (M, g) be a contact CR-submanifold of of a Sasaki-like almost contact manifold with B-metric (M̄, ϕ, η, ζ, ḡ, g˜¯). then the radical distribution Rad(TM) of radical transversal lightlike submanifold (M, g̃) is integrable and the shape operatorAξ*of the screen distribution of (M, g̃) vanishes identically on Rad(TM), for any ξ ∈ Rad(TM).