Seon-Bu Kim

Volume Comparison Theorems for Lorentzian Manifolds

Using Riccati equation techniques and the Raychaudhuri equation from General Relativity, volume comparison results are obtained for compact geodesic wedges in the chronological future of some point in a globally hyperbolic space-time and corresponding wedges in a Lorentzian space-form.

A NOTE ON CHEN'S BASIC EQUALITY FOR SUBMANIFOLDS IN A SASAKIAN SPACE FORM

It is proved that a Riemannian manifold M isometrically immersed in a Sasakian space form M ˜ ( c ) of constant φ -sectional curvature c 1 , with the structure vector field ξ tangent to M , satisfies Chens basic equality if and only if it is a 3 -dimensional minimal invariant submanifold.

CONFORMAL VECTOR FIELDS AND TOTALLY UMBILIC HYPERSURFACES

In this article, we show that if a semi-Riemannian space form carries a conformal vector field V of which the tangential part on a connected hypersurface ecomes a conformal vector field and the normal part on does not vanish identically, then is totally umbilic. Furthermore, we give a complete description of conformal vector fields on semi-Riemannian space forms.

A focal index theorem for null geodesics

Abstract Employing techniques recently developed by D. Kalish for Riemannian manifolds, we obtain a focal Morse index theorem for a null geodesic segment initially and terminally perpendicular to spacelike submanifolds of arbitrary codimension in a general space-time.

A basic inequality for submanifolds in locally conformal almost cosymplectic manifolds

For submanifolds tangent to the structure vector field in locally conformal almost cosymplectic manifolds of pointwise constantφ-sectional curvature, we establish a basic inequality between the main intrinsic invariants of the submanifold on one side, namely its sectional curvature and its scalar curvature; and its main extrinsic invariant on the other side, namely its squared mean curvature. Some applications including inequalities between the intrinsic invariantδM and the squared mean curvature are given. The equality cases are also discussed.

Nonisolated spacelike focal and conjugate points in Lorentzian geometry

Nonisolated focal and conjugate points along a spacelike geodesic are investigated. We give examples of nonisolated focal points and provide conditions for such phenomenon motivated by Helfer’s example of nonisolated conjugate points and Kupeli’s definitions of degenerate focal and conjugate points.