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 $L^{\pm }(\mathbb {S}^{n+1}_{p,1})$ of a pseudo-Riemannian space form $\mathbb {S}^{n+1}_{p,1}$ of non-vanishing curvature enjoys a Kähler or para-Kähler structure $(\mathbb {J},\mathbb {G})$ which is in addition Einstein. Moreover, in the three-dimensional case, $L^{\pm }(\mathbb {S}^{n+1}_{p,1})$ enjoys another Kähler or para-Kähler structure $(\mathbb {J}’,\mathbb {G}’)$ which is scalar flat. The normal congruence of a hypersurface $\mathcal {S}$ of $\mathbb {S}^{n+1}_{p,1}$ is a Lagrangian submanifold $\bar {\mathcal {S}}$ of $L^{\pm }(\mathbb {S}^{n+1}_{p,1})$, and we relate the local geometries of $\mathcal {S}$ and $\bar {\mathcal {S}}.$ In particular $\bar {\mathcal {S}}$ is totally geodesic if and only if $\mathcal {S}$ has parallel second fundamental form. In the three-dimensional case, we prove that $\bar {\mathcal {S}}$ is minimal with respect to the Einstein metric $\mathbb {G}$ (resp. with respect to the scalar flat metric $\mathbb {G}’$) if and only if it is the normal congruence of a minimal surface $\mathcal {S}$ (resp. of a surface $\mathcal {S}$ with parallel second fundamental form); moreover $\bar {\mathcal {S}}$ is flat if and only if $\mathcal {S}$ is Weingarten.