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On a variational problem for curves in Lie sphere geometry

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Let \(\Lambda \) be the unit tangent bundle of the unit 3-sphere acted on transitively by the contact group of Lie sphere transformations. We study the Lie sphere geometry of generic curves in \(\Lambda \) which are everywhere transversal to the contact distribution of \(\Lambda \). By the method of moving frames, we prove that such curves can be parametrized by a Lie-invariant parameter, the Lie arclength, and that in this parametrization they are uniquely determined, up to Lie sphere transformation, by four local invariants, the Lie curvatures. We then consider the simplest Lie-invariant functional on generic transversal curves defined by integrating the differential of the Lie arclength. The corresponding Euler–Lagrange equations are computed and the critical curves are characterized in terms of their Lie curvatures. In our discussion, we adopt Griffiths’ exterior differential systems approach to the calculus of variations.

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Data Availibility Statement

No datasets were generated or analysed during the current study.

Notes

  1. To be specified in Section 3.

  2. See (2.9) in Section 2.

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Acknowledgements

The author gratefully acknowledges valuable input by Emilio Musso. This work was partially supported by PRIN 2017 and 2022 “Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics" (protocolli 2017JZ2SW5-004 and 2022AP8HZ9-003); and by GNSAGA of INdAM. The author would like to thank the referee for his comments and suggestions.

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  1. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124, Parma, Italy

    Lorenzo Nicolodi

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  1. Lorenzo Nicolodi
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L.N. wrote and reviewed the manuscript.

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Correspondence to Lorenzo Nicolodi.

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In memory of Professor Lieven Vanhecke (1939-2023).

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Nicolodi, L. On a variational problem for curves in Lie sphere geometry. Ann Glob Anal Geom 68, 18 (2025). https://doi.org/10.1007/s10455-025-10021-4

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