Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions
HTML articles powered by AMS MathViewer
- by J. A. Jaramillo, M. Jiménez-Sevilla and L. Sánchez-González PDF
- Proc. Amer. Math. Soc. 142 (2014), 1075-1087 Request permission
Abstract:
In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete $C^{k}$ Finsler manifold $M$ is determined by the normed algebra $C_b^k(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete $C^{k}$ Finsler manifold $M$ is determined by the algebra $C_b^k(M)$; (ii) the weak Finsler structure of a separable and complete $C^{k}$ Finsler manifold $M$ modeled on a Banach space with a Lipschitz and $C^k$ smooth bump function is determined by the algebra $C^k_b(M)$; (iii) the weak Finsler structure of a $C^1$ uniformly bumpable and complete $C^{1}$ Finsler manifold $M$ modeled on a Weakly Compactly Generated (WCG) Banach space is determined by the algebra $C^1_b(M)$; and (iv) the isometric structure of a WCG Banach space $X$ with a $C^1$ smooth bump function is determined by the algebra $C_b^1(X)$.References
- Daniel Azagra, Juan Ferrera, and Fernando López-Mesas, Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220 (2005), no. 2, 304–361. MR 2119282, DOI 10.1016/j.jfa.2004.10.008
- Daniel Azagra, Javier Gómez Gil, Jesús A. Jaramillo, Mauricio Lovo, and Robb Fry, $C^1$-fine approximation of functions on Banach spaces with unconditional basis, Q. J. Math. 56 (2005), no. 1, 13–20. MR 2124575, DOI 10.1093/qmath/hah020
- D. Azagra, J. Ferrera, F. López-Mesas, and Y. Rangel, Smooth approximation of Lipschitz functions on Riemannian manifolds, J. Math. Anal. Appl. 326 (2007), no. 2, 1370–1378. MR 2280987, DOI 10.1016/j.jmaa.2006.03.088
- M. Bachir and G. Lancien, On the composition of differentiable functions, Canad. Math. Bull. 46 (2003), no. 4, 481–494. MR 2011388, DOI 10.4153/CMB-2003-047-2
- 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
- Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
- Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
- Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176, DOI 10.1007/978-1-4757-3480-5
- M. Isabel Garrido, Jesús A. Jaramillo, and Ángeles Prieto, Banach-Stone theorems for Banach manifolds, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 (2000), no. 4, 525–528. Perspectives in mathematical analysis (Spanish). MR 1865754
- M. Isabel Garrido and Jesús A. Jaramillo, Variations on the Banach-Stone theorem, Extracta Math. 17 (2002), no. 3, 351–383. IV Course on Banach Spaces and Operators (Spanish) (Laredo, 2001). MR 1995413, DOI 10.1081/amp-120005381
- M. I. Garrido and J. A. Jaramillo, Homomorphisms on function lattices, Monatsh. Math. 141 (2004), no. 2, 127–146. MR 2037989, DOI 10.1007/s00605-002-0011-4
- Isabel Garrido, Jesús A. Jaramillo, and Yenny C. Rangel, Algebras of differentiable functions on Riemannian manifolds, Bull. Lond. Math. Soc. 41 (2009), no. 6, 993–1001. MR 2575330, DOI 10.1112/blms/bdp077
- Isabel Garrido, Olivia Gutú, and Jesús A. Jaramillo, Global inversion and covering maps on length spaces, Nonlinear Anal. 73 (2010), no. 5, 1364–1374. MR 2661232, DOI 10.1016/j.na.2010.04.069
- I. Garrido, J. A. Jaramillo, and Y. C. Rangel, Lip-density and algebras of Lipschitz functions on metric spaces, Extracta Math. 25 (2010), no. 3, 249–261. MR 2857997
- Joaquín M. Gutiérrez and José G. Llavona, Composition operators between algebras of differentiable functions, Trans. Amer. Math. Soc. 338 (1993), no. 2, 769–782. MR 1116313, DOI 10.1090/S0002-9947-1993-1116313-5
- Petr Hájek and Michal Johanis, Smooth approximations, J. Funct. Anal. 259 (2010), no. 3, 561–582. MR 2644097, DOI 10.1016/j.jfa.2010.04.020
- J. R. Isbell, Algebras of uniformly continuous functions, Ann. of Math. (2) 68 (1958), 96–125. MR 103407, DOI 10.2307/1970045
- M. Jiménez-Sevilla and L. Sánchez-González, On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler manifolds, Nonlinear Anal. 74 (2011), no. 11, 3487–3500. MR 2803076, DOI 10.1016/j.na.2011.03.004
- Serge Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, vol. 191, Springer-Verlag, New York, 1999. MR 1666820, DOI 10.1007/978-1-4612-0541-8
- Nicole Moulis, Approximation de fonctions différentiables sur certains espaces de Banach, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 4, 293–345 (French, with English summary). MR 375379, DOI 10.5802/aif.400
- 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
- S. B. Myers, Algebras of differentiable functions, Proc. Amer. Math. Soc. 5 (1954), 917–922. MR 65823, DOI 10.1090/S0002-9939-1954-0065823-1
- Mitsuru Nakai, Algebras of some differentiable functions on Riemannian manifolds, Jpn. J. Math. 29 (1959), 60–67. MR 120587, DOI 10.4099/jjm1924.29.0_{6}0
- Karl-Hermann Neeb, A Cartan-Hadamard theorem for Banach-Finsler manifolds, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), 2002, pp. 115–156. MR 1950888, DOI 10.1023/A:1021221029301
- Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, DOI 10.1016/0040-9383(66)90013-9
- Y. C. Rangel, Algebras de funciones diferenciables en variedades, Ph.D. Dissertation (Departmento de Analisis Matematico, Facultad de Matemáticas, Universidad Complutense de Madrid), 2008.
- Patrick J. Rabier, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. of Math. (2) 146 (1997), no. 3, 647–691. MR 1491449, DOI 10.2307/2952457
Additional Information
- J. A. Jaramillo
- Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Email: jaramil@mat.ucm.es
- M. Jiménez-Sevilla
- Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Email: marjim@mat.ucm.es
- L. Sánchez-González
- Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Email: lfsanche@mat.ucm.es
- Received by editor(s): September 1, 2011
- Received by editor(s) in revised form: April 23, 2012
- Published electronically: December 17, 2013
- Additional Notes: The third author was supported by grant MEC AP2007-00868
This work was supported in part by DGES (Spain) Project MTM2009-07848 - Communicated by: Thomas Schlumprecht
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1075-1087
- MSC (2010): Primary 58B10, 58B20, 46T05, 46T20, 46E25, 46B20, 54C35
- DOI: https://doi.org/10.1090/S0002-9939-2013-11834-4
- MathSciNet review: 3148541