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On the Geometry of the Orbits of Hermann Actions

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Abstract

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes of the orbits and a new proof of a Weyl-type integration formula for Hermann actions.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, 103 MSTB, Irvine, CA, 92697-3875, USA

    Oliver Goertsches

  2. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Köln, Germany

    Gudlaugur Thorbergsson

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  1. Oliver Goertsches
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  2. Gudlaugur Thorbergsson
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Correspondence to Oliver Goertsches.

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Goertsches, O., Thorbergsson, G. On the Geometry of the Orbits of Hermann Actions. Geom Dedicata 129, 101–118 (2007). https://doi.org/10.1007/s10711-007-9198-9

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  • DOI: https://doi.org/10.1007/s10711-007-9198-9

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