Lightlike hypersurfaces of an indefinite kaehler manifold with an (ℓ, m)-type connection

Abstract

Jin [1] defined an (ℓ, m)-type connection on semi-Riemannian manifolds. Semi-symmetric non-metric connection and non-metric φ-symmetric connection are two important examples of this connection such that (ℓ, m) = (1, 0) and (ℓ, m) = (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 (ℓ, m)-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 (ℓ, m)-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 (ℓ, m)-type connection. With a semi-symmetric non-metric connection, we show that an indefinite complex space form is flat. ? 2020 by authors, all rights reserved.