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A note on Carnot geodesics in nilpotent Lie groups

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Abstract

We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct a group, with a left invariant bracket-generating distribution, for which some Carnot geodecics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, which we derive (for any Lie group) explicitly in terms of the structure constants.

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Author notes

  1. Chr Golé

    Present address: Department of Mathematics, SUNY at Stony Brook, 11794, Stony Brook, NY, USA

  2. R. Karidi

    Present address: Department of Mathematics, Stanford University, 94305, Stanford, CA, USA

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    1. Chr Golé
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    2. R. Karidi
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    Additional information

    The first author was partially supported by an NSF Postdoctoral Fellowship, and the second author was partially supported by a Rothschild Postdoctoral Fellowship.

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    Golé, C., Karidi, R. A note on Carnot geodesics in nilpotent Lie groups. Journal of Dynamical and Control Systems 1, 535–549 (1995). https://doi.org/10.1007/BF02255895

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    • DOI: https://doi.org/10.1007/BF02255895

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