Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces

TL;DR

This paper explores the geometric structures of geodesic spaces in pseudo-Riemannian space forms, relating hypersurface geometry to their normal congruences, and identifies conditions for minimality and flatness of these congruences.

Contribution

It introduces natural K"ahler and para-K"ahler structures on geodesic spaces and links hypersurface properties to Lagrangian submanifolds in these spaces.

Findings

The space of geodesics admits Einstein and scalar flat K"ahler or para-K"ahler structures.

Normal congruences are Lagrangian submanifolds with properties tied to hypersurface geometry.

Conditions for minimality and flatness of normal congruences are established.

Abstract

We describe natural K\"ahler or para-K\"ahler structures of the spaces of geodesics of pseudo-Riemannian space forms and relate the local geometry of hypersurfaces of space forms to that of their normal congruences, or Gauss maps, which are Lagrangian submanifolds. The space of geodesics L(S^{n+1}_{p,1}) of a pseudo-Riemannian space form S^{n+1}_{p,1} of non-vanishing curvature enjoys a K\"ahler or para-K\"ahler structure (J,G) which is in addition Einstein. Moreover, in the three-dimensional case, L(S^{n+1}_{p,1}) enjoys another K\"ahler or para-K\"ahler structure (J',G') which is scalar flat. The normal congruence of a hypersurface s of S^{n+1}_{p,1} is a Lagrangian submanifold \bar{s} of L(S^{n+1}_{p,1}), and we relate the local geometries of s and \bar{s}. In particular \bar{s} is totally geodesic if and only if s has parallel second fundamental form. In the three-dimensional case, we prove that \bar{s} is minimal with respect to the Einstein metric G (resp. with respect to the scalar flat metric G') if and only if it is the normal congruence of a minimal surface s (resp. of a surface s with parallel second fundamental form); moreover \bar{s} is flat if and only if s is Weingarten.