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

Abstract

We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones $$\Lambda _{c}(p)$$ of de Sitter and anti-de Sitter spacetimes. Every light cone $$\Lambda _{c}(p)$$ contains the lightlike geodesics starting from p and, essentially, it coincides with the horisms $$E^{+}(p)\cup E^{-}(p)$$ . 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.