The Myers-Steenrod theorem for Finsler manifolds of low regularity

Matveev, Vladimir; Troyanov, Marc

doi:10.1090/proc/13407

The Myers-Steenrod theorem for Finsler manifolds of low regularity
HTML articles powered by AMS MathViewer

by Vladimir S. Matveev and Marc Troyanov

Proc. Amer. Math. Soc. 145 (2017), 2699-2712

DOI: https://doi.org/10.1090/proc/13407

Published electronically: February 10, 2017

PDF | Request permission

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.

References

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, DOI 10.1007/s10474-013-0381-1
F. Brickell, On the differentiability of affine and projective transformations, Proc. Amer. Math. Soc. 16 (1965), 567–574. MR 178430, DOI 10.1090/S0002-9939-1965-0178430-4
Salomon Bochner and Deane Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639–653. MR 18187, DOI 10.2307/1969226
Eugenio Calabi and Philip Hartman, On the smoothness of isometries, Duke Math. J. 37 (1970), 741–750. MR 283727
Paul Centore, Volume forms in Finsler spaces, Houston J. Math. 25 (1999), no. 4, 625–640. MR 1829124
Gyula Csató, Bernard Dacorogna, and Olivier Kneuss, The pullback equation for differential forms, Progress in Nonlinear Differential Equations and their Applications, vol. 83, Birkhäuser/Springer, New York, 2012. MR 2883631, DOI 10.1007/978-0-8176-8313-9
Shaoqiang Deng and Zixin Hou, The group of isometries of a Finsler space, Pacific J. Math. 207 (2002), no. 1, 149–155. MR 1974469, DOI 10.2140/pjm.2002.207.149
Shaoqiang Deng, Homogeneous Finsler spaces, Springer Monographs in Mathematics, Springer, New York, 2012. MR 2962626, DOI 10.1007/978-1-4614-4244-8
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
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
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
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
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
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, DOI 10.4310/MRL.2014.v21.n2.a11
Alexander Lytchak and Asli Yaman, On Hölder continuous Riemannian and Finsler metrics, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2917–2926. MR 2216252, DOI 10.1090/S0002-9947-06-04195-X
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, DOI 10.1070/RM1997v052n02ABEH001792
Gaven J. Martin, The Hilbert-Smith conjecture for quasiconformal actions, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 66–70. MR 1694197, DOI 10.1090/S1079-6762-99-00062-1
Vladimir S. Matveev and Marc Troyanov, The Binet-Legendre metric in Finsler geometry, Geom. Topol. 16 (2012), no. 4, 2135–2170. MR 3033515, DOI 10.2140/gt.2012.16.2135
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, DOI 10.1007/s40879-015-0046-4
A. A. George Michael, On locally Lipschitz locally compact transformation groups of manifolds, Arch. Math. (Brno) 43 (2007), no. 3, 159–162. MR 2354804
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
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, DOI 10.2307/1968928
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
John Pardon, The Hilbert-Smith conjecture for three-manifolds, J. Amer. Math. Soc. 26 (2013), no. 3, 879–899. MR 3037790, DOI 10.1090/S0894-0347-2013-00766-3
Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
Du an Repov and Evgenij c̆epin, A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Ann. 308 (1997), no. 2, 361–364. MR 1464908, DOI 10.1007/s002080050080
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
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, DOI 10.1007/BF00975178
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
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, DOI 10.1007/BF02675081
S. Z. Shefel′, Smoothness of a conformal mapping of Riemannian spaces, Sibirsk. Mat. Zh. 23 (1982), no. 1, 153–159, 222 (Russian). MR 651886
Michael Taylor, Existence and regularity of isometries, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2415–2423. MR 2204038, DOI 10.1090/S0002-9947-06-04090-6