Rolling Stiefel Manifolds Equipped with α -Metrics

Markus Schlarb, Knut Hüper, Irina Markina and Fátima Silva Leite ()
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Markus Schlarb: Institute of Mathematics, Julius-Maximilians-Universität Würzburg, 97074 Würzburg, Germany
Knut Hüper: Institute of Mathematics, Julius-Maximilians-Universität Würzburg, 97074 Würzburg, Germany
Irina Markina: Department of Mathematics, University of Bergen, P.O. Box 7803, N-5020 Bergen, Norway
Fátima Silva Leite: Institute of Systems and Robotics, University of Coimbra, Pólo II, 3030-290 Coimbra, Portugal

Mathematics, 2023, vol. 11, issue 21, 1-36

Abstract: We discuss the rolling, without slipping and without twisting, of Stiefel manifolds equipped with α -metrics, from an intrinsic and an extrinsic point of view. We, however, start with a more general perspective, namely, by investigating the intrinsic rolling of normal naturally reductive homogeneous spaces. This gives evidence as to why a seemingly straightforward generalization of the intrinsic rolling of symmetric spaces to normal naturally reductive homogeneous spaces is not possible, in general. For a given control curve, we derive a system of explicit time-variant ODEs whose solutions describe the desired rolling. These findings are applied to obtain the intrinsic rolling of Stiefel manifolds, which is then extended to an extrinsic one. Moreover, explicit solutions of the kinematic equations are obtained, provided that the development curve is the projection of a not necessarily horizontal one-parameter subgroup. In addition, our results are put into perspective with examples of the rolling Stiefel manifolds known from the literature.

Keywords: intrinsic rolling; extrinsic rolling; Stiefel manifolds; normal naturally reductive homogeneous spaces; covariant derivatives; parallel vector fields; kinematic equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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