Journal of the Mathematical Society of Japan

The asymptotic cones of manifolds of roughly non-negative radial curvature

Yukihiro MASHIKO, Koichi NAGANO, and Kazuo OTSUKA

Full-text: Open access

Abstract

We prove that the asymptotic cone of every complete, connected, non-compact Riemannian manifold of roughly non-negative radial curvature exists, and it is isometric to the Euclidean cone over their Tits ideal boundaries.

Article information

Source
J. Math. Soc. Japan, Volume 57, Number 1 (2005), 55-68.

Dates
First available in Project Euclid: 13 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1160745813

Digital Object Identifier
doi:10.2969/jmsj/1160745813

Mathematical Reviews number (MathSciNet)
MR2114720

Zentralblatt MATH identifier
1075.53028

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

Keywords
comparison geometry asymptotic cones Tits ideal boundary Alexandrov spaces

Citation

MASHIKO, Yukihiro; NAGANO, Koichi; OTSUKA, Kazuo. The asymptotic cones of manifolds of roughly non-negative radial curvature. J. Math. Soc. Japan 57 (2005), no. 1, 55--68. doi:10.2969/jmsj/1160745813. https://projecteuclid.org/euclid.jmsj/1160745813


Export citation

References

  • U. Abresch, Lower curvature bounds, Toponogov's theorem and bounded topology I, Ann. Sci. École Norm. Sup., 28 (1985), 651–670.
  • A.,D. Alexandrov, V.,N. Berestovskii and I.,G. Nikolaev, Generalized Riemannian spaces, Uspekhi Mat. Nauk, 41 (1986), no.,3, 3–44; translation in Russian Math. Surveys, 41 (1986), no.,3, 1–54.
  • A.,D. Alexandrov and V.,A. Zalgaller, Intrinsic geometry of surfaces, Translations of Math. Monographs, vol.,15, Amer. Math. Soc., 1967.
  • W. Ballmann, Lectures on spaces of nonpositive curvature, DMV-seminar, Band 25, Birkhäuser, Basel-Boston-Berlin, 1995.
  • W. Ballmann, M. Gromov and V. Schroeder, Manifolds of nonpositive curvature, Progr. Math., 61, Birkhäuser, Boston-Basel-Stuttgart, 1985.
  • R.,L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1–49.
  • D. Burago, Yu. Burago and S. Ivanov, A course in metric geometry, Grad. Stud. Math., Volume 33, Amer. Math. Soc., 2001.
  • Yu. Burago, M. Gromov and G. Perel'man, A.,D. Alexandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk, 47 (1992), no.,2, 3–51, 222; translation in Russian Math. Surveys, 47 (1992), no.,2, 1–58.
  • J. Cheeger, Critical points of distance functions and applications to geometry, Geometric topology: recent developments, (Montecatini Terme, 1990), Lecture Notes in Math., 1504, Springer, Berlin, 1991, 1–38.
  • G. Drees, Asymptotically flat manifolds of non-negative curvature, Differential Geom. Appl., 4 (1994), 77–90.
  • M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv., 56, no.,2, (1981), 179–195.
  • M. Gromov, rédigé par J. Lafontaine et P. Pansu, Structures métriques pour les variétés riemanniennes, Cedic/Fernand Nathan, 1981.
  • M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, (eds. J. Lafontaine and P. Pansu), Progr. Math., 152, Birkhäuser, Boston-Basel-Berlin, 1998.
  • L. Guijarro and V. Kapovitch, Restrictions on the geometry at infinity of nonnegatively curved manifolds, Duke Math. J., 78, no.,2, (1995), 257–276.
  • 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, Tohoku Math. Publ., 20, Tohoku Univ., Sendai, 2001, 61–68.
  • Y. Itokawa, Y. Machigashira and K. Shiohama, Generalized Toponogov's Theorem for manifolds with radial curvature bounded below, Exploration in complex and Riemannian geometry, a volume dedicated to Robert E. Greene, Contemp. Math., Vol.,332 (2003), 121–130.
  • A. Kasue, A compactification of a manifolds with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup., 21, no.,4, (1988), 593–622.
  • J. Lott and Z. Shen, Manifolds with quadratic curvature decay and slow volume growth, Ann. Sci. École Norm. Sup., 33, no.,4, (2000), 275–290.
  • Y. Machigashira, Complete open manifolds of non-negative radial curvature, Pacific J. Math, 165, no.,1, (1994), 153–160.
  • Y. Machigashira, Manifolds of roughly non-negative radial curvature, preprint, 2001.
  • Y. Machigashira and K. Shiohama, Riemannian manifolds with positive radial curvature, Japan. J. Math., no.,2, (1993), 419–430.
  • V.,B. Marenich and S.,J. Mendonça, Manifolds with minimal radial curvature bounded from below and big radius, Indiana Univ. Math. J., 48, no.,1, (1999), 249–274.
  • K. Nagano and K. Otsuka, The asymptotic cones of manifolds with asymptotically non-negative curvature, preprint (unpublished), 2001.
  • A. Petrunin and W. Tuschmann, Asymptotic flatness and cone structure at infinity, Math. Ann., 321, no.,4, (2001), 775–788.
  • K. Shiohama, T. Shioya and M. Tanaka, The geometry of total curvature on complete open surfaces, Cambridge Tracts in Math., 159, Cambridge Univ. Press, 2003.
  • T. Shioya, The ideal boundaries of complete open surfaces, Tohoku Math. J., 43, no.,1, (1991), 37–59.
  • T. Shioya, Mass of rays in Alexandrov spaces of nonnegative curvature, Comment. Math. Helv., 69, no.,2, (1994), 208–228.
  • Y.,T. Siu and S.,T. Yau, Complete Kahler manifolds with nonpositive curvature of faster than quadratic decay, Ann. of Math. (2), 105 (1977), no.,2, 225–264.
  • T. Yamaguchi, A convergence theorem in the geometry of Alexandrov space, Actes de la Table Ronde de Géométrie Différentielle, Luminy, 1992, Sémin. Congr., 1, Soc. Math. France, Paris, 1996, 601–642.
  • T. Yamaguchi, Isometry groups of spaces with curvature bounded above, Math. Z., 232 (1999), no.,2, 275–286.
  • Y.,H. Yang, Some remarks on manifolds with asymptotically nonnegative sectional curvature, Kobe J. Math., 15 (1998), 157–164.
  • S.,H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications, Amer. J. Math., 116 (1994), 669–682.