Dae Ho Jin

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

We study lightlike hypersurfaces M of an indefinite Kaehler manifold M̅ of quasi-constant curvature subject to the condition that the characteristic vector field ζ of M̅ is tangent to M. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface M of M̅ such that (1) the screen distribution S(TM) is totally umbilical or (2) M is screen conformal.

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces.

LIGHTLIKE HYPERSURFACES OF AN INDEFINITE KAEHLER MANIFOLD WITH A NON-METRIC 𝜙-SYMMETRIC CONNECTION non-metric ${\phi}$-symmetric connection;lightlike hypersurface;recurrent and Lie-recurrent hypersurfaces; We define a new connection on semi-Riemannian manifold, which is called a non-metric ${\phi}$-symmetric connection. Semi-symmetric non-metric connection and quarter-symmetric non-metric connection are two impotent examples of this connection. The purpose of this paper is to study the geometry of lightlike hypersurfaces of an indefinite Kaehler manifold with a non-metric ${\phi}$-symmetric connection.

In this paper, we give the explicit determinations of dual plane curves, general dual helices and dual slant helices in terms of its dual curvature and dual torsion as a fundamental theory of dual curves in a dual 3-space

We study the geometry of half lightlike submanifolds 𝑀 of an indefinite cosymplectic manifold 𝑀. First, we construct two types of half lightlike submanifolds according to the form of the structure vector field of 𝑀, named by tangential and ascreen half lightlike submanifolds. Next, we characterize the lightlike geometries of such two types of half lightlike submanifolds.