Luigi Verdiani

Totally Geodesic Orbits of Isometries

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non-semisimple Lie group with an hypersurface orbit.

Complexn-dimensional manifolds with a realn2-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n2-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dimℝ Aut (M) ≽ (dimℂM)2.

Invariant Metrics on Cohomogeneity One Manifolds

Let G be one of the connected subgroups of the orthogonal group of ℝn which acts transitively on the unit sphere Sn−1. We get the necessary and sufficient condition for G-invariant metrics g on ℝn\{0} to be extendend to the origin. For n=2 this is a classical result of Berard–Bergery. The curvature tensor and the sectional curvature of any such Riemannian G-manifold (ℝn, g) are described in terms of the length of the Killing vector fields, as well as the second fundamental form of the regular orbits G(P)=Sn−1. As an application we describe all G-invariant metrics which are Kähler, hyperKähler or have constant principal curvatures. Some of these results are generalized to the case of any cohomogeneity one G-manifold which, in a neighbourhood of a singular orbit, can be identified with a twisted product.

Seven dimensional cohomogeneity one manifolds with nonnegative curvature

We show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature, unless it admits one with positive curvature. As a consequence, the classification of nonnegatively curved cohomogeneity one manifolds in dimension seven is reduced to only one further family of candidates.