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

Abstract

We describe natural Kähler or para-Kähler 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 Kähler or para-Kähler structure (J,G) which is in addition Einstein. Moreover, in the three-dimensional case, L(S^{n+1}_{p,1}) enjoys another Kähler or para-Kähler 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.

30 pages

References in corpus (5)

• On the Geometry of Spaces of Oriented Geodesics
• A characterization of Weingarten surfaces in hyperbolic 3-space
• Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface
• On maximal surfaces in the space of oriented geodesics of hyperbolic 3-space
• Isometric Immersions of the Hyperbolic Plane into the Hyperbolic Space