Invariant Metrics on Cohomogeneity One Manifolds |
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Authors: | Luigi Verdiani |
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Institution: | (1) Dipartimento di Matematica Applicata, Università degli Studi di Firenze, Via S. Marta 3, 50134 Fierenze, Italy |
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Abstract: | Let G be one of the connected subgroups of the orthogonal group of
n
which acts transitively on the unit sphere S
n–1. We get the necessary and sufficient condition for G-invariant metrics g on
n
\{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 (
n
, 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)=S
n–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. |
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Keywords: | cohomogeneity one isometric actions |
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