A Note on Spacelike Submanifolds Through Light Cones in Lorentzian Space Forms

We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones Lambda(c)(p) of de Sitter and anti-de Sitter space-times. Every light cone Lambda(c)(p) contains the lightlike geodesics starting from p and, essentially, it coincides with the horisms E+(p) boolean OR E- (p). The analysis works by means of an explicit correspondence with the spacelike submanifolds through the light cone in the Lorentz-Minkowski spacetime. In particular, a characterization of totally umbilical compact surfaces through light cones in de Sitter and anti-de Sitter is shown and we obtain an estimation of the first eigenvalue of the Laplace operator on a compact spacelike surface in a light cone.