Generalized space forms

Authors:
Neil N. Katz and Kei Kondo

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2279-2284

MSC (2000):
Primary 53C21; Secondary 53C20

DOI:
https://doi.org/10.1090/S0002-9947-02-02966-5

Published electronically:
February 14, 2002

MathSciNet review:
1885652

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Abstract | References | Similar Articles | Additional Information

Abstract: Spaces with radially symmetric curvature at base point are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to or under a radially symmetric metric, to with Riemannian universal covering of equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity radius at .

**1.**U. Abresch,*Lower curvature bounds, Toponogov's theorem and bounded topology I*, Ann. Sci. Ecole Norm. Sup.,**19**(1985) 651-670. MR**87j:53058****2.**U. Abresch,*Lower curvature bounds, Toponogov's theorem and bounded topology II*, Ann. Sci. Ecole Norm. Sup.,**20**(1987) 475-502. MR**89d:53080****3.**A. Allamigeon,*Propriétés globales des espaces de Riemann harmoniques*, Ann. Inst. Fourier,**15**(1965) 91-132. MR**33:6549****4.**A.L. Besse,*Manifolds all of whose Geodesics are Closed*, Springer-Verlag, Berlin-Heidelberg, 1978. MR**80c:53044****5.**J. Cheeger,*Critical Points of Distance Functions and Applications to Geometry*, in*Geometric Topology: Recent Developments*, Lecture Notes in Math.**1504**, Springer-Verlag, Berlin-Heidelberg, 1991. MR**94a:53075****6.**D. Elerath,*An improved Toponogov comparison theorem for non-negatively curved manifolds*, J. Differential Geometry,**15**(1980) 187-216. MR**83b:53039****7.**R.E. Greene and H. Wu,*Function Theory on Manifolds which Possess a Pole*, Lecture Notes in Math.**699**, Springer-Verlag, Berlin-Heidelberg, 1979. MR**81a:53002****8.**K. Grove,*Critical Point Theory for Distance Functions*, Proc. of Symposia in Pure Math.,**54**Part 3, Amer. Math. Soc., Providence, RI, 1993. MR**94f:53065****9.**Y. Itokawa, Y. Machigashira and K. Shiohama,*Generalized Toponogov's theorem for manifolds with radial curvature bounded below*, preprint.**10.**Y. Machigashira and K. Shiohama,*Riemannian manifolds with positive radial curvature*, Japan. J. Math.,**19**(1994) 419-430. MR**95f:53080****11.**V. Marenich,*Manifolds with minimal radial curvature bounded from below and big volume*, Trans. Amer. Math. Soc.,**352**(2000) 4451-4468.**12.**F.W. Warner,*Conjugate loci of constant order*, Ann. of Math.,**86**(1967) 192-212. MR**35:4857**

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Additional Information

**Neil N. Katz**

Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Honjoh 1, Saga 840-8502, Japan

Email:
katz@ms.saga-u.ac.jp

**Kei Kondo**

Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Honjoh 1, Saga 840-8502, Japan

Email:
kondok@ms.saga-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-02-02966-5

Keywords:
Radial curvature,
rigidity

Received by editor(s):
June 12, 2001

Received by editor(s) in revised form:
September 27, 2001

Published electronically:
February 14, 2002

Additional Notes:
The first author was supported by the Japan Society for the Promotion of Science and Monbusho Grant-in-Aid of Research No. 13099720.

Article copyright:
© Copyright 2002
American Mathematical Society