We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones
of de Sitter and anti-de Sitter spacetimes 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 rst eigenvalue
of the Laplace operator on a compact spacelike surface in a light cone
