Polar actions on compact Euclidean hypersurfaces |
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Authors: | Ion Moutinho Ruy Tojeiro |
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Affiliation: | 1. Universidade Federal Fluminense, Niteroi, 24020-140, Brazil
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Abstract: | Given an isometric immersion of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso 0(M n ) of the isometry group Iso(M n ) of M n admits an orthogonal representation such that for every . If G is a closed connected subgroup of Iso(M n ) acting polarly on M n , we prove that Φ(G) acts polarly on , and we obtain that f(M n ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure. |
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Keywords: | Polar actions Rotation hypersurfaces Isoparametric submanifolds Rigidity of hypersurfaces Warped products |
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