Geometric invariants and focal surfaces of spacelike curves in de Sitter space from a caustic viewpoint

Abstract

The focal surface of a generic space curve in Euclidean $ 3 $-space is a classical subject which is a two dimensional caustic and has Lagrangian singularities. In this paper, we define the first de Sitter focal surface and the second de Sitter focal surface of de Sitter spacelike curve and consider their singular points as an application of the theory of caustics and Legendrian dualities. The main results state that de Sitter focal surfaces can be seen as two dimensional caustics which have Lagrangian singularities. To characterize these singularities, a useful new geometric invariant $ \rho(s) $ is discovered and two dual relationships between focal surfaces and spacelike curve are given. Three examples are used to demonstrate the main results.