Tohoku Mathematical Journal

Comparison geometry referred to warped product models

Yukihiro Mashiko and Katsuhiro Shiohama

Full-text: Open access

Abstract

We generalize the Alexandrov-Toponogov comparison theorem to the case of complete Riemannian manifolds referred to warped product models. We prove the maximal diameter theorem and the rigidity theorem. In particular, we discuss collapsing phenomena where the curvature explosion may occur.

Article information

Source
Tohoku Math. J. (2), Volume 58, Number 4 (2006), 461-473.

Dates
First available in Project Euclid: 1 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1170347684

Digital Object Identifier
doi:10.2748/tmj/1170347684

Mathematical Reviews number (MathSciNet)
MR2297194

Zentralblatt MATH identifier
1135.53023

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

Keywords
Riemannian manifolds geodesics radial sectional curvature Hausdorff convergence

Citation

Mashiko, Yukihiro; Shiohama, Katsuhiro. Comparison geometry referred to warped product models. Tohoku Math. J. (2) 58 (2006), no. 4, 461--473. doi:10.2748/tmj/1170347684. https://projecteuclid.org/euclid.tmj/1170347684


Export citation

References

  • U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology, Ann. Sci. École Norm. Sup. (4) 18 (1985), 651--670.
  • U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology II, Ann. Sci. École Norm. Sup. (4) 20 (1987), 475--502.
  • M. Berger, Les variétés riemanniennes $(1/4)$-pincées, Ann. Scuola Norm. Sup. Pisa (3) 14 (1960), 161--170.
  • M. Berger, Sur les variétés à courbure positive de diamétre minimum, Comment. Math. Helv. 35 (1961), 28--34.
  • M. E. Gage, Upper bounds for the first eigenvalue of the Laplace-Beltrami operator, Indiana Univ. Math. J. 29 (1980), 897--912.
  • R. Greene and H. C. Wu, Function theory on manifolds which possess a pole, Lecture Notes Math. 699, Springer, Berlin, 1979.
  • E. Heintze and H. Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. (4) 11 (1978), 451--470.
  • Y. Itokawa, Y. Machigashira and K. Shiohama, Generalized Toponogov's theorem for manifolds with radial curvature bounded below, Exploration in complex and Riemannian geometry, 121--130, Contemp. Math. 332, Amer. Math. Soc., Providence, R.I., 2003.
  • Y. Itokawa, Y. Machigashira and K. Shiohama, Maximal diameter theorems for manifolds with restricted radial curvature, Proceedings of the Fifth Pacific Rim Geometry Conference (Sendai, 2000), 61--68, Tohoku Math. Publ. 20, Tohoku Univ., Sendai, 2001.
  • W. Klingenberg, Manifolds with restricted conjugate locus, Ann. of Math. (2) 78 (1963), 527--547.
  • N. Katz and K. Kondo, Generalized space forms, Trans. Amer. Math. Soc. 354 (2002), 2279--2284.
  • K. Kondo, The topology of complete manifolds with radial curvature bounded below, Preprint, Saga University, 2002.
  • Y. Machigashira, Manifolds with pinched radial curvature, Proc. Amer. Math. Soc. 118 (1993), 979--985.
  • Y. Machigashira, Complete open manifolds of non-negative radial curvature, Pacific J. Math. 165 (1994), 153--160.
  • Y. Machigashira and K. Shiohama, Riemannian manifolds with positive radial curvature, Japan. J. Math. (N. S.) 19 (1993), 419--430.
  • M. Maeda, Volume estimate of submanifolds in compact Riemannian manifolds, J. Math. Soc. Japan 30 (1978), 533--551.
  • Y. Mashiko, K. Nagano and K. Otsuka, The asymptotic cones of manifolds of roughly non-negative radial curvature, J. Math. Soc. Japan 57 (2005), 55--68.
  • Y. Mashiko and K. Shiohama, The axiom of plane for warped product models and its application, Kyushu J. Math. 59 (2005), 385--392.
  • H.Omori, A class of Riemannian metrics on a manifold, J. Differential Geometry 2 (1968), 233--252.
  • Y. Otsu, Topology of complete open manifolds with nonnegative Ricci curvature, Geometry of manifolds (Matsumoto, 1988), 295--302, Perspect. Math. 8, Academic Press, Boston, M.A., 1990.
  • T. Sakai, On Riemannian manifolds admitting a function whose gradient is of constant norm, Kodai Math. J. 19 (1996), 39--51.
  • T. Sakai, Warped products and Riemannian manifolds admitting a function whose gradient is of constant norm, Math. J. Okayama Univ. 39 (1997), 165--185.
  • T. Sakai, On Riemannian manifolds admitting a function whose gradient is of constant norm II, Kodai Math. J. 21 (1998), 102--124.
  • T. Shioya, The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature, Trans. Amer. Math. Soc. 351 (1999), 1765--1801.
  • K. Shiohama and M. Tanaka, Compactification and maximal diameter theorem for noncompact manifolds with radial curvaturebounded below, Math. Z. 241 (2002), 341--351.
  • M. Tanaka, On the cut loci of a von Mangoldt's surface of revolution, J. Math. Soc. Japan 44 (1992), 631--641.
  • S. H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvatureand its applications, Amer. J. Math. 116 (1994), 669--682.