2012/09/26 by Antonio Caminha, Caminha, Antonio
Mathematics · #53C42 (Primary) 53C30 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C30 #msc:53C42
paper · pdf · doi:10.48550/arxiv.1209.5946
In this version we also treat spacelike hypersurfaces of Lorentzian groups
arxiv created 2014/01/01 · arxiv updated 2014/01/03
In this paper, we study the geometry of a connected oriented cmc Riemannian hypersurface M of a semi-Riemannian group G of Lie algebra \mathfrak g and index 0 or 1. If G is Riemannian and M is compact and transversal to an element of \mathfrak g, we show that it is a lateral class of a closed embedded Lie subgroup of G; we also do this if G is Lorentzian, provided M has sufficiently large mean curvature. If G is Riemannian semisimple and M is compact, we prove that M has degenerate Gauss map and minimal relative nullity at least 1. We also extend the above results to the case where M is complete and noncompact. For a Riemannian G, we show that a minimal M is either transversal to an element of \mathfrak g, hence stable, or has degenerate Gauss map and minimal relative nullity at least 1; for M cmc and transversal to an element of \mathfrak g, if we ask the immersion to be proper and have bounded second fundamental form, then M is also a lateral class of a closed embedded Lie subgroup of G, provided a certain growing condition on the size of the corresponding Gauss map is satisfied. Finally, for a Lorentzian group G, with sectional curvatures bounded from above on Lorentzian planes, we extend a result of Y. Xin, proving that a complete M is totally umbilical, provided it is transversal to a timelike element of \mathfrak g, has large enough mean curvature and bounded hyperbolic Gauss map.