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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

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


Author: Henri Anciaux
Journal: Trans. Amer. Math. Soc. 366 (2014), 2699-2718
MSC (2010): Primary 53C50, 53C25, 53D12, 53C42
Published electronically: September 25, 2013
MathSciNet review: 3165652
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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.


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Additional Information

Henri Anciaux
Affiliation: Department of Mathematics, University of Sao Paulo, Sao Paulo 05508-090, Brazil
Email: henri.anciaux@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05972-7
Received by editor(s): December 8, 2011
Received by editor(s) in revised form: September 10, 2012
Published electronically: September 25, 2013
Additional Notes: The author was supported by CNPq (PQ 302584/2007-2) and Fapesp (2010/18752-0)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.