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Stability of Left-Invariant Totally Geodesic Unit Vector Fields on Three-Dimensional Lie Groups

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Geometry and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 72))

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Abstract

We consider the problem of stability or instability of unit vector fields on three-dimensional Lie groups with left-invariant metric which have totally geodesic image in the unit tangent bundle with the Sasaki metric with respect to classical variations of volume. We prove that among non-flat groups only SO(3) of constant curvature + 1 admits stable totally geodesic submanifolds of this kind. Restricting the variations to left-invariant (i.e., equidistant) ones, we give a complete list of groups which admit stable/unstable unit vector fields with totally geodesic image.

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Acknowledgements

The author thanks Prof. Vladimir Rovenski (Haifa University, Israel) for hospitality during the 2-nd International Workshop on Geometry and Symbolic Computations (Haifa, May 15–20, 2013) and the MAPLE developers for the perfect tool in complicated calculations. The author also thanks Ye. Petrov (Kharkiv National University) for careful reading of preliminary version and helpful suggestions.

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Authors and Affiliations

  1. Department of Geometry, Faculty of Mechanics and Mathematics, Kharkiv Karazin National University, Svobody Sq. 4, 61022, Kharkiv, Ukraine

    Alexander Yampolsky

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  1. Alexander Yampolsky
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Correspondence to Alexander Yampolsky .

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Editors and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, Israel

    Vladimir Rovenski

  2. Faculty of Mathematics, University of Łód´z', Łód´z', Poland

    Paweł Walczak

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Yampolsky, A. (2014). Stability of Left-Invariant Totally Geodesic Unit Vector Fields on Three-Dimensional Lie Groups. In: Rovenski, V., Walczak, P. (eds) Geometry and its Applications. Springer Proceedings in Mathematics & Statistics, vol 72. Springer, Cham. https://doi.org/10.1007/978-3-319-04675-4_8

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