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St. Petersburg Mathematical Journal

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Differentiation in metric spaces


Author: A. Lytchak
Original publication: Algebra i Analiz, tom 16 (2004), nomer 6.
Journal: St. Petersburg Math. J. 16 (2005), 1017-1041
MSC (2000): Primary 53C20
DOI: https://doi.org/10.1090/S1061-0022-05-00888-5
Published electronically: November 22, 2005
MathSciNet review: 2117451
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Abstract | References | Similar Articles | Additional Information

Abstract: Differentiation of Lipschitz maps between abstract metric spaces is discussed. Differentiability of isometries, first variation formula, and Rademacher-type theorems are studied.


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  • [BBI01] D. Burago, Yu. Burago, and S. Ivanov, A course in metric geometry, Grad. Stud. Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001. MR 1835418 (2002e:53053)
  • [Bel96] A. Bellaiche, The tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1-78. MR 1421822 (98a:53108)
  • [Ber87] V. N. Berestovskii, ``Submetries'' of three-dimensional forms of nonnegative curvature, Sibirsk. Mat. Zh. 28 (1987), no. 4, 44-56; English transl., Siberian Math. J. 28 (1987), no. 4, 552-562. MR 0906032 (89h:53078)
  • [BG00] V. Berestovskii and L. Guijarro, A metric characterization of Riemannian submersions, Ann. Global Anal. Geom. 18 (2000), no. 6, 577-588. MR 1800594 (2002b:53041)
  • [BGP92] Yu. Burago, M. Gromov, and G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2, 3-51; English transl., Russian Math. Surveys 47 (1992), no. 2, 1-58. MR 1185284 (93m:53035)
  • [BH99] M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • [CH70] E. Calabi and Ph. Hartman, On the smoothness of isometries, Duke Math. J. 37 (1970), 741-750. MR 0283727 (44:957)
  • [Che99] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. MR 1708448 (2000g:53043)
  • [Fed59] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. MR 0110078 (22:961)
  • [Hal00] S. Halbeisen, On tangent cones of Alexandrov spaces with curvature bounded below, Manuscripta Math. 103 (2000), no. 2, 169-182. MR 1796313 (2001g:53078)
  • [HM98] K. Hofmann and S. Morris, The structure of compact groups, de Gruyter Stud. Math., vol. 25, Walter de Gruyter and Co., Berlin, 1998. MR 1646190 (99k:22001)
  • [JL01] W. Johnson and L. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, North-Holland, Amsterdam, 2001, pp. 1-84. MR 1863689 (2003f:46013)
  • [Kir94] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113-123. MR 1189747 (94g:28013)
  • [KL97] B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. No. 86 (1997), 115-197 (1998). MR 1608566 (98m:53068)
  • [Lyta] A. Lytchak, Almost convex subsets (in preparation).
  • [Lytb] -, Differentiation in Carnot-Carathéodory spaces (in preparation).
  • [Lytc] -, Open map theorem in metric spaces, Preprint.
  • [LY] A. Lytchak and A. Yaman, On Hölder continuous Riemannian and Finsler manifolds, Trans. Amer. Math. Soc. (to appear).
  • [Mit85] J. Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), no. 1, 35-45. MR 0806700 (87d:53086)
  • [MM00] G. A. Margulis and G. D. Mostow, Some remarks on the definition of tangent cones in a Carnot-Carathéodory space, J. Anal. Math 80 (2000), 299-317. MR 1771529 (2001d:53033)
  • [Nik95] I. Nikolaev, The tangent cone of an Aleksandrov space of curvature $ \le k$, Manuscripta Math. 86 (1995), 137-147. MR 1317739 (95m:53062)
  • [OT] Y. Otsu and H. Tanoue, The Riemannian structure of Alexandrov spaces with curvature bounded above, Preprint.
  • [Pet94] A. Petrunin, Applications of quasigeodesics and gradient curves, Comparison Geometry (Berkeley, CA, 1993-94), Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203-219. MR 1452875 (98m:53061)
  • [PP94a] G. Ya. Perel'man and A. M. Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, Algebra i Analiz 5 (1993), no. 1, 242-256; English transl., St. Petersburg Math. J. 5 (1994), no. 1, 215-227. MR 1220499 (94h:53055)
  • [PP94b] -, Quasigeodesics and gradient curves in Alexandrov spaces, Preprint, 1994.
  • [Res93] Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, Geometry, 4, Itogi Nauki i Tekhniki Ser. Sovrem. Probl. Mat. Fund. Naprav., vol. 70, VINITI, Moscow, 1989, pp. 7-189; English transl., Encyclopaedia Math. Sci., vol. 70, Springer, Berlin, 1993, pp. 3-163. MR 1099202 (92b:53104); MR 1263964

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Additional Information

A. Lytchak
Affiliation: Mathematisches Institut, Universität Bonn, Beringstr. 1, Bonn 53115, Germany
Email: lytchak@math.uni-bonn.de

DOI: https://doi.org/10.1090/S1061-0022-05-00888-5
Keywords: Aleksandrov spaces, Rademacher theorem, variation formulas, tangent cones
Received by editor(s): May 12, 2004
Published electronically: November 22, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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