On Geodesic Paracontact CR-lightlike Submanifolds

In present paper we investigate geodesic paracontact CR-lightlike submanifolds of para-Sasakian manifolds. Also some geometric results on paracontact screen CR-lightlike submanifolds are given.

On a semi-Riemannian manifold M 2n+1 , S. Kaneyuki, M. Konzai [11] introduced a structure which is called the almost paracontact structure and then they characterized the almost paracomplex structure on M 2n+1 × R. Recently, S. Zamkovoy [12] studied paracontact metric manifolds and some subclasses which is known para-Sasakian manifolds. The study of paracontact geometry has been continued by several papers ( [13,14,15,16,17]) which are contained role of paracontact geometry about semi-Riemannian geometry, mathematical physics and relationships with the para-Kähler manifolds.
In this manuscript we study the lightlike submanifolds of para-Sasakian manifolds and obtain several geometric results. The paper arranged as follow. In Section 2, we recall some basic facts about lightlike submanifolds and almost paracontact metric manifolds, respectively. In Section 3, we introduce geodesic paracontact CR-lightlike submanifolds of para-Sasakian manifolds and obtain some necessary and sufficient conditions for totally geodesic,D−geodesic,D−geodesic and mixed geodesic paracontact CR-lightlike submanifolds. Section 4 is devoted to paracontact screen CRlightlike submanifolds of para-Sasakian manifolds. By use of (2.3), the Gauss and Weingarten formulas are given bȳ ).∇ and ∇ t are linear connections on M and on the vector bundle tr(T M ), respectively.
In view of (2.2), we consider the projection morphisms L and S of tr(T M ) on ltr(T M ) and S(T M ⊥ ). Therefore (2.5) and (2.6) becomē Assume that P is a projection of T M on S(T M ), then by use of (2.1), we have Using (2.10) and (2.11), we getḡ In genereal, the induced connection ∇ on M is not metric connection. By using property of∇ and (2.7), we have However, ∇ * is a metric connection on S(T M ).

Proof. By using the definition, M isD−geodesic if and only if
So, from (2.7), (2.10) with (2.20), we obtain which completes the proof.  Proof. We know that M isD−geodesic if and only if So, in view of (2.7) and (2.9), we obtain The proof follows from (3.7) and (3.8). Proof. Assume that M is mixed geodesic. Therefore we can statē By use of (2.7), (2.9) with (2.20), we obtain The proof follows from (3.9) and (3.10). Thus one can state following decompositions:

Geodesic Paracontact Screen CR-light like Submanifolds
Let us denoteD =D⊥{ξ}. For any X tangent to M, we put where P X ∈ Γ(D) and F X ∈ Γ(φD ⊥ ).
Similarly, for any W ∈ Γ(S(T M ⊥ )), we put where BW ∈ Γ(D ⊥ ) and CW ∈ Γ(S(T M ⊥ ) −φD ⊥ ). Proof. Assume that M is totally geodesic. Then we get By use of property of Lie derivative, we havē Similarly, we get Hence the proof is completed.