# Polar Actions on Berger Spheres

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- World Scientific Publishing
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Polar actions on Berger spheres. Antonio J. Di Scala Post print (i.e. final draft post-refereeing) version of an article published on Int. J. Geom. Methods Mod. Phys. 3 (2006), no. 5-6, 1019-1023., DOI: 10.1142/S0219887806001466 . Beyond the journal formatting, please note that there could be minor changes from this document to the final published version. The final published version is accessible from here: http://www.worldscientific.com/doi/abs/10.1142/S0219887806001466 This document has made accessible through PORTO, the Open Access Repository of Politecnico di Torino (http://porto. polito.it), in compliance with the Publisher’s copyright policy as reported in the SHERPA-ROMEO website: http://www.sherpa.ac.uk/romeo/issn/0219-8878/ Dedicated to Dmitri V. Alekseevsky on the occasion of his sixty-fifth birthday. Abstract The object of this article is to study a torus action on a so called Berger sphere. We also make some comments on polar actions on naturally reductive homogeneous spaces. Finally, we prove a rigidity-type theorem for Rieman- nian manifolds carrying a polar action with a fix point. Mathematics Subject Classification(2000): 53C40, 53C35 . Keywords: polar actions, Killing vector fields, totally geodesic section, Berger spheres. 1 Introduction. In a generic Riemannian manifold (M, g) there are neither Killing vector fields (i.e. infinitesimal isometries) nor non trivial totally geodesic submanifolds. The concept of polar actions is a good example where both objects come together nicely. A Lie subgroup of isometries G ⊂ I(M, g) acts polarly on (M, g) if there exists a con- nected closed submanifold Σ meeting all G -orbits orthogonally. Notice that the sections Σ are totally geodesic submanifolds of (M, g) . So, both Killing vector fields and totally geodesic submanifolds fit together in the setting of polar actions. Polar actions have been considered by several authors, see for example [6],[10], [1]. Let G act polarly on (M, g) and let g˜ be another

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