Showing papers by "Jurgen Berndt published in 2010"

Hyperpolar homogeneous foliations on symmetric spaces of noncompact type

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 [1], 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 = SLr+1(R)/SOr+1. 1 Isometric actions on the real hyperbolic plane The special linear group G = SL2(R) = {( a b c d )∣∣∣∣ a, b, c, d ∈ R, ad− bc = 1} acts on the upper half plane H = {z ∈ C | =(z) > 0} by linear fractional transformations of the form ( a b c d ) · z = az + b cz + d . By equipping H with the Riemannian metric ds = dx + dy y2 = dzdz =(z)2 (z = x+ iy)

Cohomogeneity one actions on symmetric spaces of noncompact type

Abstract: An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As a consequence, we find many new examples of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type. We apply our conceptual approach to derive explicit classifications of cohomogeneity one actions on some symmetric spaces.