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# One-parameter family of Clairaut-Liouville metrics

Abstract : Riemannian metrics with singularities are considered on the $2$-sphere of revolution. The analysis of such singularities is motivated by examples stemming from mechanics and related to projections of higher dimensional (regular) sub-Riemannian distributions. An unfolding of the metrics in the form of an homotopy from the canonical metric on $\SS^2$ is defined which allows to analyze the singular case as a limit of standard Riemannian ones. A bifurcation of the conjugate locus for points on the singularity is finally exhibited.
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https://hal.archives-ouvertes.fr/hal-00177686
Contributor : Jean-Baptiste Caillau <>
Submitted on : Monday, February 4, 2008 - 10:41:16 PM
Last modification on : Friday, January 10, 2020 - 9:09:10 PM
Document(s) archivé(s) le : Friday, November 25, 2016 - 8:13:32 PM

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• HAL Id : hal-00177686, version 2

### Citation

Bernard Bonnard, Jean-Baptiste Caillau, Minoru Tanaka. One-parameter family of Clairaut-Liouville metrics. 2007. ⟨hal-00177686v2⟩

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