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The Myers-Steenrod theorem for Finsler manifolds of low regularity


Authors: Vladimir S. Matveev and Marc Troyanov
Journal: Proc. Amer. Math. Soc. 145 (2017), 2699-2712
MSC (2010): Primary 53B40, 53C60, 35B65
DOI: https://doi.org/10.1090/proc/13407
Published electronically: February 10, 2017
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Abstract: We prove a version of Myers-Steenrod's theorem for Finsler manifolds under the minimal regularity hypothesis. In particular we show that an isometry between $ C^{k,\alpha }$-smooth (or partially smooth) Finsler metrics, with $ k+\alpha >0$, $ k\in \mathbb{N} \cup \{0\}$, and $ 0 \leq \alpha \leq 1$ is necessarily a diffeomorphism of class $ C^{k+1,\alpha }$. A generalization of this result to the case of Finsler 1-quasiconformal mapping is given. The proofs are based on the reduction of the Finslerian problems to Riemannian ones with the help of the Binet-Legendre metric.


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  • [1] B. Aradi and D. Cs. Kertész, Isometries, submetries and distance coordinates on Finsler manifolds, Acta Math. Hungar. 143 (2014), no. 2, 337-350. MR 3233537, https://doi.org/10.1007/s10474-013-0381-1
  • [2] F. Brickell, On the differentiability of affine and projective transformations, Proc. Amer. Math. Soc. 16 (1965), 567-574. MR 0178430
  • [3] Salomon Bochner and Deane Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639-653. MR 0018187
  • [4] Eugenio Calabi and Philip Hartman, On the smoothness of isometries, Duke Math. J. 37 (1970), 741-750. MR 0283727
  • [5] Paul Centore, Volume forms in Finsler spaces, Houston J. Math. 25 (1999), no. 4, 625-640. MR 1829124
  • [6] Gyula Csató, Bernard Dacorogna, and Olivier Kneuss, The pullback equation for differential forms, Progress in Nonlinear Differential Equations and their Applications, 83, Birkhäuser/Springer, New York, 2012. MR 2883631
  • [7] Shaoqiang Deng and Zixin Hou, The group of isometries of a Finsler space, Pacific J. Math. 207 (2002), no. 1, 149-155. MR 1974469, https://doi.org/10.2140/pjm.2002.207.149
  • [8] Shaoqiang Deng, Homogeneous Finsler spaces, Springer Monographs in Mathematics, Springer, New York, 2012. MR 2962626
  • [9] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
  • [10] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
  • [11] Tadeusz Iwaniec, Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions, Dissertationes Math. (Rozprawy Mat.) 198 (1982), 45. MR 670148
  • [12] Tadeusz Iwaniec and Gaven Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. MR 1859913
  • [13] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR 0152974
  • [14] Tony Liimatainen and Mikko Salo, $ n$-harmonic coordinates and the regularity of conformal mappings, Math. Res. Lett. 21 (2014), no. 2, 341-361. MR 3247061, https://doi.org/10.4310/MRL.2014.v21.n2.a11
  • [15] Alexander Lytchak and Asli Yaman, On Hölder continuous Riemannian and Finsler metrics, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2917-2926 (electronic). MR 2216252, https://doi.org/10.1090/S0002-9947-06-04195-X
  • [16] Ĭozhe Maleshich, The Hilbert-Smith conjecture for Hölder actions, Uspekhi Mat. Nauk 52 (1997), no. 2(314), 173-174 (Russian); English transl., Russian Math. Surveys 52 (1997), no. 2, 407-408. MR 1480156, https://doi.org/10.1070/RM1997v052n02ABEH001792
  • [17] Gaven J. Martin, The Hilbert-Smith conjecture for quasiconformal actions, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 66-70 (electronic). MR 1694197, https://doi.org/10.1090/S1079-6762-99-00062-1
  • [18] Vladimir S. Matveev and Marc Troyanov, The Binet-Legendre metric in Finsler geometry, Geom. Topol. 16 (2012), no. 4, 2135-2170. MR 3033515, https://doi.org/10.2140/gt.2012.16.2135
  • [19] Vladimir S. Matveev and Marc Troyanov, Completeness and incompleteness of the Binet-Legendre metric, Eur. J. Math. 1 (2015), no. 3, 483-502. MR 3401902, https://doi.org/10.1007/s40879-015-0046-4
  • [20] A. A. George Michael, On locally Lipschitz locally compact transformation groups of manifolds, Arch. Math. (Brno) 43 (2007), no. 3, 159-162. MR 2354804
  • [21] Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
  • [22] S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. (2) 40 (1939), no. 2, 400-416. MR 1503467, https://doi.org/10.2307/1968928
  • [23] I. G. Nikolaev and S. Z. Shefel, Differential properties of mappings that are conformal at a point, Sibirsk. Mat. Zh. 27 (1986), no. 1, 132-142, 199 (Russian). MR 847421
  • [24] John Pardon, The Hilbert-Smith conjecture for three-manifolds, J. Amer. Math. Soc. 26 (2013), no. 3, 879-899. MR 3037790, https://doi.org/10.1090/S0894-0347-2013-00766-3
  • [25] Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
  • [26] Dusan Repovs and Evgenij Scepin, A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Ann. 308 (1997), no. 2, 361-364. MR 1464908, https://doi.org/10.1007/s002080050080
  • [27] Ju. G. Rešetnjak, Differential properties of quasiconformal mappings and conformal mappings of Riemannian spaces, Sibirsk. Mat. Zh. 19 (1978), no. 5, 1166-1183, 1216 (Russian). MR 508507
  • [28] I. Kh. Sabitov, On the smoothness of isometries, Sibirsk. Mat. Zh. 34 (1993), no. 4, 169-176, iv, x (Russian, with English and Russian summaries); English transl., Siberian Math. J. 34 (1993), no. 4, 741-748. MR 1248802, https://doi.org/10.1007/BF00975178
  • [29] I. Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics, Reviews in Mathematics and Mathematical Physics, vol. 13, Cambridge Scientific Publishers, Cambridge, 2008. MR 2584444
  • [30] E. V. Shchepin, Hausdorff dimension and the dynamics of diffeomorphisms, Mat. Zametki 65 (1999), no. 3, 457-463 (Russian, with Russian summary); English transl., Math. Notes 65 (1999), no. 3-4, 381-385. MR 1717521, https://doi.org/10.1007/BF02675081
  • [31] S. Z. Shefel, Smoothness of a conformal mapping of Riemannian spaces, Sibirsk. Mat. Zh. 23 (1982), no. 1, 153-159, 222 (Russian). MR 651886
  • [32] Michael Taylor, Existence and regularity of isometries, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2415-2423 (electronic). MR 2204038, https://doi.org/10.1090/S0002-9947-06-04090-6

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Additional Information

Vladimir S. Matveev
Affiliation: Institut für Mathematik, Friedrich-Schiller Universität Jena, 07737 Jena, Germany
Email: vladimir.matveev@uni-jena.de

Marc Troyanov
Affiliation: Section de Mathématiques, École Polytechnique Féderale de Lausanne, station 8, 1015 Lausanne, Switzerland
Email: marc.troyanov@epfl.ch

DOI: https://doi.org/10.1090/proc/13407
Keywords: Finsler metric, isometries, Myers-Steenrod theorem, Binet-Legendre metric
Received by editor(s): May 12, 2016
Received by editor(s) in revised form: July 27, 2016
Published electronically: February 10, 2017
Additional Notes: The authors thank the Friedrich-Schiller-Universität Jena, EPFL and the Swiss National Science Foundation for their support.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society

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