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A quadrangle comparison theorem and its application to soul theory for Alexandrov spaces |
Jianguo CAO1,2, Bo DAI3( ), Jiaqiang MEI2,4 |
| 1. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA; 2. Department of Mathematics, Nanjing University, Nanjing 210093, China; 3. LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China; 4. Institute of Mathematical Science, Nanjing University, Nanjing 210093, China |
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Abstract We shall derive two sufficient conditions for complete finitedimensional Alexandrov spaces of nonnegative curvature to be contractible. One of the new technical tools used in our proof is a quadrangle comparison theorem inspired by Perelman.
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| Keywords
Alexandrov space with nonnegative curvature
soul theory
quadrangle comparison theorem
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Corresponding Author(s):
DAI Bo,Email:daibo@math.pku.edu.cn
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Issue Date: 01 February 2011
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| 1 |
Burago D, Burage Yu, Ivanov S. A Course in Metric Geometry.Providence: Amer Math Soc, 2001
|
| 2 |
Burago Yu, Gromov M, Perelman G. A.D. Alexandrov spaces with curvature bounded below. Russian Mathematical Surveys, 1992, 47: 1-58
doi: 10.1070/RM1992v047n02ABEH000877
|
| 3 |
Cao J, Dai B, Mei J. An optimal extension of Perelman’s comparison theorem for quadrangles and its applications. In: Lee Yng-Ing, Lin Chang-Shou, Tsui Mao-Pei, eds. Recent Advances in Geometric Analysis. Advanced Lectures in Mathematics, Vol 11. Beijing and Boston: Higher Education Press and International Press, 2009-2010, 39-59
|
| 4 |
Cheeger J, Gromoll D. On the structure of complete manifolds of nonnegative curvature. Annals of Mathematics, 1972, 96: 413-443
doi: 10.2307/1970819
|
| 5 |
Greene R. A genealogy of noncompact manifolds of nonnegative curvature: history and logic. In: Grove K, Petersen P, eds. Comparison Geometry. MSRI Publications Vol 30. New York: Cambridge University Press, 1997, 99-134
|
| 6 |
Gromoll D, Meyer W. On complete open manifolds of positive curvature. Annals of Mathematics , 1969, 90: 75-90
doi: 10.2307/1970682
|
| 7 |
Kapovitch V, Petrunin A, Tuschmann W. Nilpotency, almost nonnegative curvature and the gradient push. Annals of Mathematics, 2010, 171(1): 343-373
doi: 10.4007/annals.2010.171.343
|
| 8 |
Perelman G. Alexandrov’s spaces with curvatures bounded from below II. 1991, http://www.math.psu.edu/petrunin/papers/papers.html
|
| 9 |
Perelman G. Proof of the soul conjecture of Cheeger and Gromoll. Journal of Differential Geometry, 1994, 40: 209-212
|
| 10 |
Perelman G. Spaces with curvature bounded below. In: Proceedings of the International Congress of Mathematicians, Vol 1, 2 (Zürich, 1994). Basel: Birkh?user Verlag, 1995, 517-525
|
| 11 |
Perelman G, Petrunin A. Extremal subsets in Alexandrov spaces and the generalized Liebman theorem. St Petersburg Mathematical Journal, 1994, 5(1): 215-227
|
| 12 |
Petrunin A. Semiconcave functions in Alexandrov’s geometry. In: Cheeger J, Grove K, eds. Surveys in Differential Geometry, Volume XI, Metric and Comparison Geometry. Sommerville: International Press, 2007, 137-201 ; See also http://www.math.psu.edu /petrunin/papers/papers.html
|
| 13 |
Sharafutdinov V. The Pogorelov-Klingenberg theorem for manifolds that are homeomorphic to ?n. Siberian Mathematical Journal, 1977, 18(4): 649-657
doi: 10.1007/BF00967208
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