Generalized Legendre Transform of Conformally Flat Metrics
In the calculus of variations, an important role is played by the Minkowski duality, or the Legendre transform of convex functions. We consider weakly regular, conformally flat Riemannian metrics of nonnegative curvature defined on the n -dimensional unit sphere. For this class of metrics, an analog...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 277; no. 5; pp. 760 - 769 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.12.2023
Springer Nature B.V |
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| Abstract | In the calculus of variations, an important role is played by the Minkowski duality, or the Legendre transform of convex functions. We consider weakly regular, conformally flat Riemannian metrics of nonnegative curvature defined on the
n
-dimensional unit sphere. For this class of metrics, an analog of the Legendre transformation is introduced and studied in detail. |
|---|---|
| AbstractList | In the calculus of variations, an important role is played by the Minkowski duality, or the Legendre transform of convex functions. We consider weakly regular, conformally flat Riemannian metrics of nonnegative curvature defined on the
n
-dimensional unit sphere. For this class of metrics, an analog of the Legendre transformation is introduced and studied in detail. In the calculus of variations, an important role is played by the Minkowski duality, or the Legendre transform of convex functions. We consider weakly regular, conformally flat Riemannian metrics of nonnegative curvature defined on the n-dimensional unit sphere. For this class of metrics, an analog of the Legendre transformation is introduced and studied in detail. |
| Author | Semenov, S. P. Kurkina, M. V. Rodionov, E. D. Slavsky, V. V. |
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| Cites_doi | 10.1007/s10958-007-0472-z 10.1007/BF02104826 |
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| Copyright | Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Keywords | 53A07 Lobachevsky space Legendre transform conformally flat metric |
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| References | M. V. Kurkina, S. P. Semenov, and V. V. Slavskii, “Numerical implementation of the Legendre transform of conformally convex function,” in: Proc. Int. Conf. “Lomonosov Readings on Altai. Fundamental Problems of Science and Technology” (Barnaul, Novermber 13-16, 2018) [in Russian], Altai State Univ., Barnaul (2018), pp. 321–326. V. V. Slavskii, “Conformally flat metrics of bounded curvature on the n-dimensional sphere,” in: Studies on Geometry “in the whole” and Mathematical Analysis [in Russian], Nauka, Novosibirsk (1987), pp. 183–199. V. V. Balashchenko, Yu. G. Nikonorov, E. D. Rodionov, and V. V. Slavskii, Homogeneous Spaces: Theory and Applications, Poligrafist, Khanty-Mansiysk (2008). Yu. G. Reshetnyak, Stability Theorems in Geometry and Analysis [in Russian], Novosibirsk (1996). ToponogovVADifferential Geometry of Curves and Surfaces2012MoscowFizmatkniga[in Russian] SlavskiiVVConformally flat metrics and pseudo-Euclidean geometrySib. Mat. Zh.1994353674682129222810.1007/BF02104826 RodionovEDSlavskiiVVOne-dimensional sectional curvature of Riemannian manifoldsDokl. Ross. Akad. Nauk200238744544572006029 NikonorovYuGRodionovEDSlavskiiVVGeometry of homogeneoues Riemannian manifoldsJ. Math. Sci.2007146663136390256857210.1007/s10958-007-0472-z KurkinaMVRodionovEDSlavskiiVVConformally convex functions and conformally flat metrics of nonnegative curvatureDokl. Ross. Akad. Nauk20154622141143 MV Kurkina (6884_CR2) 2015; 462 6884_CR5 6884_CR7 ED Rodionov (6884_CR6) 2002; 387 6884_CR1 6884_CR3 VA Toponogov (6884_CR9) 2012 VV Slavskii (6884_CR8) 1994; 35 YuG Nikonorov (6884_CR4) 2007; 146 |
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| SubjectTerms | Calculus of variations Mathematical analysis Mathematics Mathematics and Statistics |
| Title | Generalized Legendre Transform of Conformally Flat Metrics |
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