A cone splitting theorem for Alexandrov spaces

Pacific Journal of Mathematics

Volumn 218, Issue 1, 2005, Pages 1-15

  Alexander, Stephanie B ^a   Bishop, Richard L ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Affine functions; Alexandrov spaces; Warped products

Indexed keywords

EID: 18444400266     PISSN: 00308730     EISSN: None     Source Type: Journal    
DOI: 10.2140/pjm.2005.218.1     Document Type: Article

References (29)

1. [Aleksandrov et al. 1986] A. D. Aleksandrov, V. N. Berestovskil, and I. G. Nikolaev, "Generalized Riemannian spaces", Russian Math. Surveys 41:3 (1986), 1-54. MR 88e:53103 Zbl 0625.53059
2. [Alexander and Bishop 1998] S. B. Alexander and R. L. Bishop, "Warped products of Hadamard spaces", Manuscripta Math. 96:4 (1998), 487-505. MR 99g:53070 Zbl 0909.53050
3. [Alexander and Bishop 2003] S. Alexander and R. L. Bishop, "ℱK-convex functions on metric spaces", Manuscripta Math. 110:1 (2003), 115-133. MR2004a:53100
4. [Alexander and Bishop ≥ 2005] S. Alexander and R. L. Bishop, "Curvature bounds for warped products of metric spaces", http://www.math.uiuc.edu/~sba/wp.pdf. To appear in Geom. and Funct. Anal.
5. [Ballmann 1995] W. Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser, Basel, 1995. MR 97a:53053
6. [Berestovskii 1983] V. N. Berestovskil, "Borsuk's problem on metrization of a polyhedron", Dokl. Akad. Nauk SSSR 268:2 (1983), 273-277.
7. English translation: Soviet Math. Dokl. 27:1 (1983), 273-277. MR85b:51017
8. [Berestovskii and Nikolaev 1993] V. N. Berestovskij and I. G. Nikolaev, "Multidimensional generalized Riemannian spaces", pp. 165-243, 245-250 in Geometry IV, edited by Y. G. Reshetnyak, Encyclopaedia Math. Sci. 70, Springer, Berlin, 1993. MR 1 263 965
9. [Bridson and Haefliger 1999] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Math. Wissenschaften 319, Springer, Berlin, 1999. MR 2000k:53038
10. [Burago et al. 1992] Y. Burago, M. Gromov, and G. Perelman, "A. D. Aleksandrov spaces with curvatures bounded below", Uspekhi Mat. Nauk 47:2(284) (1992), 3-51, 222.
11. English translation: Russian Math. Surveys 47:2 (1992), 1-58. MR 93m:53035
12. [Burago et al. 2001] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Math. 33, American Mathematical Society, Providence, RI, 2001. MR 2002e:53053 Zbl 0981.51016
13. [Buyalo 1995] S. Buyalo, "Lecture notes on spaces of nonpositive curvature", course taught at the University of Illinois Urbana-Champaign, 1995, http://www.pdmi.ras.ru/~sbuyalo/add.html.
14. [Gromov 1981] M. Gromov, "Hyperbolic manifolds, groups and actions", pp. 183-213 in Riemann surfaces and related topics, edited by I. Kra and B. Maskit, Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, 1981. MR82m:53035 Zbl 0467.53035
15. [Gromov 1999] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser, Boston, 1999. MR 2000d:53065
16. [Innami 1982] N. Innami, "Splitting theorems of Riemannian manifolds", Compositio Math. 47:3 (1982), 237-247. MR84e:53053 Zbl 0514.53040
17. [Jost. 1997] J. Jost, Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997. MR 98g:53070
18. [Kleiner 1999] B. Kleiner, "The local structure of length spaces with curvature bounded above", Math. Z. 231:3 (1999), 409-456. MR 2000m:53053
19. [Mashiko 1999] Y. Mashiko, "Affine functions on Alexandrov surfaces", Osaka J. Math. 36:4 (1999), 853-859. MR 2001a:53057 Zbl 01455440
20. [Mashiko 2002] Y. Mashiko, "A splitting theorem for Alexandrov spaces", Pacific J. Math. 204:2 (2002), 445-458. MR 2003b:53044 Zbl 01819000
21. [Milka 1967] A. D. Milka, "Metric structure of a certain class of spaces that contain straight lines", Ukrain. Geometr. Sb. Vyp. 4 (1967), 43-48. MR 41 #983
22. [Nikolaev 1995] I. Nikolaev, "The tangent cone of an Aleksandrov space of curvature ≤ K", Manuscripta Math. 86:2 (1995), 137-147. MR 95m:53062 Zbl 0822.53043
23. [Obata 1962] M. Obata, "Certain conditions for a Riemannian manifold to be iosometric with a sphere", J. Math. Soc. Japan 14 (1962), 333-340. MR 25 #5479 Zbl 0115.39302
24. [Perelman 1991] G. Perelman, "Alexandrov's spaces with curvature bounded from below, II", pre-print, 1991.
25. [Perelman and Petrunin 1994] G. Perelman and A. Petrunin, "Quasigeodesics and gradient curves in Alexandrov spaces", preprint, 1994, http://www.math.psu.edu/petrunin/papers/qg_ams.pdf.
26. [Petrunin ≥ 2005] A. Petrunin, in preparation.
27. [Plaut 2002] C. Plaut, "Metric spaces of curvature ≥ k", pp. 819-898 in Handbook of geometric topology, North-Holland, Amsterdam, 2002. MR 2002m:53063
28. [Toponogov 1959] V. A. Toponogov, "Riemannian spaces containing straight lines", Dokl. Akad. Nauk SSSR 127 (1959), 977-979.
29. English translation: "Spaces with straight lines", pp. 278-280 in Twenty-two papers on algebra, number theory and differential geometry, American Mathematical Society Translations 37, 1964. MR 21 #7520 Zbl 0138.42902