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Totally geodesic hypersurfaces of homogeneous spaces

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Abstract

We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c) the twisted product of the line and a homogeneous space (with the warping/twisting function given explicitly). In the first case, F is also a Riemannian product; in the last two cases, it is a leaf of a totally geodesic homogeneous fibration. Case (c) can alternatively be characterized by the fact that M admits a Riemannian submersion onto the universal cover of the group SL(2) equipped with a particular left-invariant metric, and F is the preimage of the two-dimensional solvable totally geodesic subgroup.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia

    Y. Nikolayevsky

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  1. Y. Nikolayevsky
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Correspondence to Y. Nikolayevsky.

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Supported by ARC Discovery grant DP130103485

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Nikolayevsky, Y. Totally geodesic hypersurfaces of homogeneous spaces. Isr. J. Math. 207, 361–375 (2015). https://doi.org/10.1007/s11856-015-1158-8

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  • DOI: https://doi.org/10.1007/s11856-015-1158-8

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