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 Mat. Sb., 2015, Volume 206, Number 5, Pages 127–160 (Mi msb8445)

Closed geodesics on piecewise smooth surfaces of revolution with constant curvature

I. V. Sypchenko, D. S. Timonina

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A theorem on the structure of breaks of generalized geodesics on piecewise smooth surfaces is established in two dimensions and $n$ dimensions. To illustrate the result, all simple closed geodesics are found: on a cylinder (with bases included), on a surface formed as a union of two spherical caps and on a surface formed as a union of two cones. In the last case the stability of the closed geodesics (in a natural finite-dimensional class of perturbations) is analysed, the conjugate points and the indices of the geodesics are found. This problem is related to finding conjugate points in piecewise smooth billiards and surfaces of revolution.
Bibliography: 40 titles.

Keywords: Riemannian geometry, piecewise smooth surface of revolution, closed geodesics, conjugate points.
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DOI: https://doi.org/10.4213/sm8445

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English version:
Sbornik: Mathematics, 2015, 206:5, 738–769

Bibliographic databases:

UDC: 514.774.8+514.76
MSC: 53A05, 53C22

Citation: I. V. Sypchenko, D. S. Timonina, “Closed geodesics on piecewise smooth surfaces of revolution with constant curvature”, Mat. Sb., 206:5 (2015), 127–160; Sb. Math., 206:5 (2015), 738–769

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8445
• https://doi.org/10.4213/sm8445
• http://mi.mathnet.ru/eng/msb/v206/i5/p127

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This publication is cited in the following articles:
1. E. A. Kudryavtseva, “Liouville integrable generalized billiard flows and Poncelet type theorems”, J. Math. Sci., 225:4 (2017), 611–638
2. R. K. Klimov, “Closed geodesics on piecewise smooth constant curvature surfaces of revolution”, Moscow University Mathematics Bulletin, 71:6 (2016), 242–247
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