| Full text | |
| Author(s): |
Anciaux, Henri
[1]
Total Authors: 1
|
| Affiliation: | [1] Univ Sao Paulo, Dept Math, BR-05508090 Sao Paulo - Brazil
Total Affiliations: 1
|
| Document type: | Journal article |
| Source: | TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY; v. 366, n. 5, p. 2699-2718, MAY 2014. |
| Web of Science Citations: | 11 |
| Abstract | |
We describe natural Kahler or para-Kahler 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-p,1(n+1)) of a pseudo-Riemannian space form S-p,1(n+1) of non-vanishing curvature enjoys a Kahler or para-Kahler structure (J, G) which is in addition Einstein. Moreover, in the three-dimensional case, L +/-(S-p,1(n+1)) enjoys another Kahler or para-Kahler structure (J', G') which is scalar flat. The normal congruence of a hypersurface S of S-p,1(n+1) is a Lagrangian submanifold (S) over barS of L +/-((n+1)(p,1)), and we relate the local geometries of S and (S) over bar. In particular (S) over bar is totally geodesic if and only if S has parallel second fundamental form. In the three-dimensional case, we prove that (S) over bar 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 (S) over bar is flat if and only if S is Weingarten. (AU) | |
| FAPESP's process: | 10/18752-0 - Lagrangian submanifolds in pseudo-Riemannian geometry |
| Grantee: | Henri Nicolas Guillaume Anciaux |
| Support Opportunities: | Regular Research Grants |