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Isometries of Carnot Groups and Sub-Finsler Homogeneous Manifolds

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Abstract

We show that isometries between open sets of Carnot groups are affine. This result generalizes a result of Hamenstädt. Our proof does not rely on her proof. We show that each isometry of a sub-Riemannian manifold is determined by the horizontal differential at one point. We then extend the result to sub-Finsler homogeneous manifolds. We discuss the regularity of isometries of homogeneous manifolds equipped with homogeneous distances that induce the manifold topology.

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Acknowledgments

Both authors would like to thank Université Paris Sud, Orsay, where part of this research was conducted. This paper has benefited from discussions with E. Breuillard and P. Pansu. Special thanks go to them. Moreover, the authors are particularly grateful to S. Nicolussi Golo and to the anonymous referee for their thorough review of the paper and their helpful remarks.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, 40014, Jyväskylä, Finland

    Enrico Le Donne

  2. CIRM Fondazione Bruno Kessler, Via Sommarive 14, 38123, Trento, Italy

    Alessandro Ottazzi

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  1. Enrico Le Donne
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  2. Alessandro Ottazzi
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Correspondence to Enrico Le Donne.

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Le Donne, E., Ottazzi, A. Isometries of Carnot Groups and Sub-Finsler Homogeneous Manifolds. J Geom Anal 26, 330–345 (2016). https://doi.org/10.1007/s12220-014-9552-8

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  • DOI: https://doi.org/10.1007/s12220-014-9552-8

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