Abstract
Jin [1] defined an ( )-type connection on semi-Riemannian manifolds. Semi-symmetric nonmetric connection and non-metric ∅-symmetric connection are two important examples of this connection such that ( ) = (1; 0) and ( ) = (0; 1), respectively. In semi-Riemannian geometry, there are few literatures for the lightlike geometry, so we expose new theories for non-degenerate submanifolds in semi-Riemannian geometry. The goal of this paper is to study a characterization of a (Lie) recurrent lightlike hypersurface M of an indefinite Kaehler manifold with an ( )-type connection when the charateristic vector field is tangnet to M. In the special case that an indefinite Kaehler manifold of constant holomorphic sectional curvature is an indefinite complex space form, we investigate a lightlike hypersurface of an indefinite complex space form with an ( )-type connection when the charateristic vector field is tangnet to M. Moreover, we show that the total space, the complex space form, is characterized by the screen conformal lightlike hypersurface with an ( )-type connection. With a semi-symmetric non-metric connection, we show that an indefinite complex space form is flat.
Highlights
A linear connection ∇ ̄ on a semi-Riemannian manifold (M, g) is called an (, m)-type connection [1] if ∇ ̄ and its torsion tensor Tsatisfy∇ ̄ X Y = ∇X Y + θ(Y ){ X + mJ X }. (1.3)The objective of study in this paper is lightlike hypersurfaces of an indefinite Kaehler manifold M = (M, g, J) with an (, m)-type connection subject to the conditions that (1) the tensor field J, defined by (1.1) and (1.2), is identical with the indefinite almost complex structure tensor J of Mand (2) the characteristic vector field ζ of Mis tangent to M
Let M be a lightlike hypersurface of an indefinite Kaehler manifold Mwith an (, m)-type connection such that ζ is tangent to M
Let M be a screen conformal lightlike hypersurface of an indefinite complex space form M (c) with an ( , m)-
Summary
A linear connection ∇ ̄ on a semi-Riemannian manifold (M , g) is called an ( , m)-type connection [1] if ∇ ̄ and its torsion tensor Tsatisfy. A complementary vector bundle S(T M ) of T M ⊥ in T M is a non-degenerate distribution on M , which is called a screen distribution on M , such that T M = T M ⊥ ⊕orth S(T M ), where ⊕orth denotes the orthogonal direct sum. We consider lightlike hypersurfaces M of an indefinite Kaehler manifold Mwith an ( , m)-type connection and a screen distribution S(T M ) which contains ζ. Let M be a lightlike hypersurface of an indefinite Kaehler manifold Mwith an ( , m)-type connection such that ζ is tangent to M.
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