Differentiation in metric spaces
Algebra i analiz, Tome 16 (2004) no. 6, pp. 128-161.

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Differentiation of Lipschitz maps between abstract metric spaces is discussed. Differentiability of isometries, first variation formula, and Rademacher-type theorems are studied.
Keywords: Alexandrov spaces, Rademacher theorem, variation formulas, tangent cones.
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A. Lytchak. Differentiation in metric spaces. Algebra i analiz, Tome 16 (2004) no. 6, pp. 128-161. http://geodesic.mathdoc.fr/item/AA_2004_16_6_a4/

[BBI01] Burago D., Burago Yu., Ivanov S., A course in metric geometry, Grad. Stud. Math., 33, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[Bel96] Bellaiche A., “The tangent space in sub-Riemannian geometry”, Sub-Riemannian Geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 1–78 | MR | Zbl

[Ber87] Berestovskii V. H., “Submetrii prostranstvennykh form neotritsatelnoi krivizny”, Sib. mat. zh., 28:4 (1987), 44–56 | MR | Zbl

[BGOO] Berestovskii V., Guijarro L., “A metric characterization of Riemannian submersions”, Ann. Global Anal. Geom., 18:6 (2000), 577–588 | DOI | MR

[BGP92] Burago Yu., Gromov M., Perelman G., “Prostranstva A. D. Aleksandrova s ogranichennymi snizu kriviznami”, Uspekhi mat. nauk, 47:2 (1992), 3–51 | MR | Zbl

[ВН99] Bridson M., Haefliger A., Metric spaces of non-positive curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, Berlin, 1999 | MR | Zbl

[CH70] Calabi E., Hartman Ph., “On the smoothness of isometries”, Duke Math. J., 37 (1970), 741–750 | DOI | MR | Zbl

[Che99] Cheeger J., “Differentiability of Lipschitz functions on metric measure spaces”, Geom. Funct. Anal., 9 (1999), 428–517 | DOI | MR | Zbl

[Fed59] Federer H., “Curvature measures”, Trans. Amer. Math. Soc., 93 (1959), 418–491 | DOI | MR | Zbl

[Hal00] Halbeisen S., “On tangent cones of Alexandrov spaces with curvature bounded below”, Manuscripta Math., 103:2 (2000), 169–182 | DOI | MR | Zbl

[HM98] Hoffmann K., Morris S., The structure of compact groups, Walter de Gruyter and Co., Berlin, 1998 | MR

[JL01] Johnson W., Lindenstrauss L., “Basic concepts in the geometry of Banach spaces”, Handbook of the Geometry of Banach Spaces, Vol. 1, North-Holland, Amsterdam, 2001, 1–84 | MR

[Kir94] Kirchheim B., “Rectifiable metric spaces: local structure and regularity of the Hausdorff measure”, Proc. Amer. Math. Soc., 121 (1994), 113–123 | DOI | MR | Zbl

[KL97] Kleiner B., Leeb B., “Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings”, Inst. Hautes Études Sci. Publ. Math., 86 (1997), 1998, 115–197 | MR

[Lyta] Lytchak A., “Almost convex subsets”, Geom. Dedicata, 115 (2005), 201–218 | DOI | MR | Zbl

[Lytb] Lytchak A., Differentiation in Carnot–Carathéodory spaces, gotovitsya k pechati

[Lytc] Lytchak A., Open map theorem in metric spaces, Preprint | MR

[LY] Lytchak A., Yaman A., “On Hölder continuous Riemannian and Finsler manifolds”, Trans. Amer. Math. Soc., 358:7 (2006), 2917–2926 | DOI | MR | Zbl

[Mit85] Mitchell J., “On Carnot–Caratheodory metrics”, J. Differential Geom., 21:1 (1985), 35–45 | MR | Zbl

[MM00] Margulis G. A., Mostow G. D., “Some remarks on the definition of tangent cones in a Carnot–Caratheodory space”, J. Anal. Math., 80 (2000), 299–317 | DOI | MR | Zbl

[Nik95] Nikolaev I., “The tangent cone of an Aleksandrov space of curvature $\leq K$”, Manuscripta Math., 86 (1995), 137–147 | DOI | MR | Zbl

[ОТ] Otsu Y., Tanoue H., The Riemannian structure of Alexandrov spaces with curvature bounded above, Preprint

[Pet94] Petrunin A., “Applications of quasigeodesics and gradient curves”, Comparison Geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997, 203–219 | MR | Zbl

[PP94a] Perelman G. Ya., Petrunin A. M., “Ekstremalnye podmnozhestva v prostranstvakh Aleksandrova i obobschennaya teorema Libermana”, Algebra i analiz, 5:1 (1993), 242–256 | MR

[PP94b] Perel'man G., Petrunin A., Quasigeodesics and gradient curves in Alexandrov spaces, Preprint, 1994 | MR

[Res93] Reshetnyak Yu. G., “Dvumernye mnogoobraziya ogranichennoi krivizny”, Geometriya – 4. Neregulyarnaya rimanova geometriya, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 70, VINITI, M., 1989, 7–189 | MR