Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Orbifolds with lower Ricci curvature bounds


Author: Joseph E. Borzellino
Journal: Proc. Amer. Math. Soc. 125 (1997), 3011-3018
MSC (1991): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-97-04046-X
MathSciNet review: 1415575
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the first betti number $b_1(O)=\dim \,H_1(O,{\mathbb R})$ of a compact Riemannian orbifold $O$ with Ricci curvature $\operatorname {Ric}(O)\ge -(n-1)k$ and diameter $\operatorname {diam}(O)\le D$ is bounded above by a constant $c(n,kD^2)\ge 0$, depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the $b_1(O)$ is bounded above by the dimension $\dim \,O$, and that if, in addition, $b_1(O)=\dim \,O$, then $O$ is a flat torus $T^n$.


References [Enhancements On Off] (What's this?)

  • [Be] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
  • [Bo] S. Bochner. Vector Fields and Ricci Curvature, Bull. Am. Math. Soc. 52 (1946), 776-797. MR 8:230a
  • [B] Joseph E. Borzellino, Orbifolds of maximal diameter, Indiana Univ. Math. J. 42 (1993), no. 1, 37–53. MR 1218706, https://doi.org/10.1512/iumj.1993.42.42004
  • [BZ] Joseph E. Borzellino and Shun-Hui Zhu, The splitting theorem for orbifolds, Illinois J. Math. 38 (1994), no. 4, 679–691. MR 1283015
  • [Br] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
  • [G] S. Gallot. A Sobolev Inequality and Some Geometric Applications, Spectra of Riemannian Manifolds, Kaigai Publications, Tokyo, 1983, 45-55.
  • [GLP] Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
  • [HD] André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1984), 70–97 (French). Transversal structure of foliations (Toulouse, 1982). MR 755163
    A. Haefliger and Quach Ngoc Du, Appendice: une présentation du groupe fondamental d’une orbifold, Astérisque 116 (1984), 98–107 (French). Transversal structure of foliations (Toulouse, 1982). MR 755164
  • [M] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR 0232311
  • [R] John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730
  • [S] A. Schwarz. A Volume Invariant of Coverings, Dokl. Ak. Nauk. USSR 105 (1955), 32-34 [in Russian].
  • [T] W. Thurston. The Geometry and Topology of 3-Manifolds, Lecture Notes, Princeton University Math. Dept., 1978.
  • [W] Joseph A. Wolf, Spaces of constant curvature, 5th ed., Publish or Perish, Inc., Houston, TX, 1984. MR 928600
  • [Z] S. Zhu. The Comparison Geometry of Ricci Curvature, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C20

Retrieve articles in all journals with MSC (1991): 53C20


Additional Information

Joseph E. Borzellino
Email: borzelli@math.psu.edu

DOI: https://doi.org/10.1090/S0002-9939-97-04046-X
Received by editor(s): May 15, 1996
Communicated by: Christopher Croke
Article copyright: © Copyright 1997 American Mathematical Society