Generalized space forms. (English) Zbl 0990.53032

Summary: Spaces with radially symmetric curvature at base point \(p\) are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to \({\mathbb R^n}\) or \(S^n\) under a radially symmetric metric, to \({\mathbb R}\text{P}^n\) with Riemannian universal covering of \(S^n\) equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity radius at \(p\).


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching
Full Text: DOI


[1] Uwe Abresch, Lower curvature bounds, Toponogov’s theorem, and bounded topology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 651 – 670. · Zbl 0595.53043
[2] U. Abresch, Lower curvature bounds, Toponogov’s theorem, and bounded topology. II, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 475 – 502. · Zbl 0651.53031
[3] André-Claude Allamigeon, Propriétés globales des espaces de Riemann harmoniques, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 2, 91 – 132 (French). · Zbl 0178.55903
[4] Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. · Zbl 0387.53010
[5] Jeff Cheeger, Critical points of distance functions and applications to geometry, Geometric topology: recent developments (Montecatini Terme, 1990) Lecture Notes in Math., vol. 1504, Springer, Berlin, 1991, pp. 1 – 38. · Zbl 0771.53015
[6] Doug Elerath, An improved Toponogov comparison theorem for nonnegatively curved manifolds, J. Differential Geom. 15 (1980), no. 2, 187 – 216 (1981). · Zbl 0526.53043
[7] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. · Zbl 0414.53043
[8] Karsten Grove, Critical point theory for distance functions, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 357 – 385. · Zbl 0806.53043
[9] Y. Itokawa, Y. Machigashira and K. Shiohama, Generalized Toponogov’s theorem for manifolds with radial curvature bounded below, preprint. · Zbl 1046.53017
[10] Yoshiroh Machigashira and Katsuhiro Shiohama, Riemannian manifolds with positive radial curvature, Japan. J. Math. (N.S.) 19 (1993), no. 2, 419 – 430. · Zbl 0804.53059
[11] V. Marenich, Manifolds with minimal radial curvature bounded from below and big volume, Trans. Amer. Math. Soc., 352 (2000) 4451-4468. · Zbl 0970.53023
[12] F. W. Warner, Conjugate loci of constant order, Ann. of Math. (2) 86 (1967), 192 – 212. · Zbl 0172.23003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.