Showing papers by "Jurgen Berndt published in 2009"
TL;DR: In this paper, the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces is studied and a characterization of the focal set in terms of its second fundamental form is provided.
Abstract: We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.
29 citations
01 Jan 2009
1 citations
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TL;DR: In this paper, the authors investigate hypersurfaces in SU 2,m, s(U_2U_m) for which C and Q are closely related to the shape of M.
Abstract: The Riemannian symmetric space SU_{2,m}/S(U_2U_m) is both Hermitian symmetric and quaternionic Kahler symmetric. Let M be a hypersurface in SU_{2,m}/S(U_2U_m) and denote by TM its tangent bundle. The complex structure of SU_{2,m}/S(U_2U_m) determines a maximal complex subbundle C of TM, and the quaternionic structure of SU_{2,m}/S(U_2U_m) determines a maximal quaternionic subbundle Q of TM. In this article we investigate hypersurfaces in SU_{2,m}/S(U_2U_m) for which C and Q are closely related to the shape of M.
1 citations
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TL;DR: A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M and admits a flat section, that is, a connected closed geodesic submanifold of M which intersects each leaf of F and intersects orthogonally at each point of intersection as mentioned in this paper.
Abstract: A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jose Carlos Diaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in arXiv:0807.3517v2 [math.DG], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SL_{r+1}(R)/SO_{r+1}.