Warped products of metric spaces of curvature bounded from above

Chen, Chien-Hsiung

doi:10.1090/S0002-9947-99-02154-6

Warped products of metric spaces of curvature bounded from above
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Trans. Amer. Math. Soc. 351 (1999), 4727-4740 Request permission

Abstract:

In this work we extend the idea of warped products, which was previously defined on smooth Riemannian manifolds, to geodesic metric spaces and prove the analogue of the theorems on spaces with curvature bounded from above.

References

Stephanie B. Alexander, I. David Berg, and Richard L. Bishop, Geometric curvature bounds in Riemannian manifolds with boundary, Trans. Amer. Math. Soc. 339 (1993), no. 2, 703–716. MR 1113693, DOI 10.1090/S0002-9947-1993-1113693-1
Stephanie B. Alexander and Richard L. Bishop, The Hadamard-Cartan theorem in locally convex metric spaces, Enseign. Math. (2) 36 (1990), no. 3-4, 309–320. MR 1096422
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
A. D. Aleksandrov, V. N. Berestovskiĭ, and I. G. Nikolaev, Generalized Riemannian spaces, Uspekhi Mat. Nauk 41 (1986), no. 3(249), 3–44, 240 (Russian). MR 854238
V. N. Berestovskii and I. G. Nikolaev, Multidimensional generalized Riemannian spaces, In book: Encyclopaedia of Math. Sciences 70, Springer, 1993, 184–242.
R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
M. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature (to appear).
Yu. D. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curvatures bounded below, Russian Math. Surveys 47:2 (1992), 1–58.
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
S. Buyalo, Lecture notes on spaces of nonpositive curvature, course taught at UIUC Spring (1995).
C. H. Chen, Warped products of metric spaces of curvature bounded from above, Trans. Amer. Math. Soc. (to appear).
M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR 624814
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246. MR 1215595
I. G. Nikolaev, Spaces of bounded curvature, lecture notes (1995), Urbana, Illinois.
Stefan Nölker, Isometric immersions of warped products, Differential Geom. Appl. 6 (1996), no. 1, 1–30. MR 1384876, DOI 10.1016/0926-2245(96)00004-6
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
Yu. G. Reshetnyak, On the theory of spaces of curvature not greater than $K$, Mat. Sb. 52 (1960), 789–798.
Yu. G. Reshetnyak, Personal communication.