Duke Mathematical Journal

A Bochner theorem and applications

Xiaochun Rong

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 91, Number 2 (1998), 381-392.

First available in Project Euclid: 19 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces


Rong, Xiaochun. A Bochner theorem and applications. Duke Math. J. 91 (1998), no. 2, 381--392. doi:10.1215/S0012-7094-98-09116-5. http://projecteuclid.org/euclid.dmj/1077232083.

Export citation


  • [Ab] U. Abrech, Uber das glatten Riemannn'scher metriken, Habilitationesschrift, Reinischen Friedrisch-Wilhelms-Universitat, Bonn, 1988.
  • [An] M. Anderson, private communication, n.d.
  • [AC] M. Anderson and J. Cheeger, $C\sp \alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 (1992), no. 2, 265–281.
  • [Ba] S. Bando, Real analyticity of solutions of Hamilton's equation, Math. Z. 195 (1987), no. 1, 93–97.
  • [BMR] J. Bemelmans, [M.] Min-Oo, and E. Ruh, Smoothing Riemannian metrics, Math. Z. 188 (1984), no. 1, 69–74.
  • [Br] G. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, vol. 46, Academic Press, New York, 1972.
  • [Ch] J. Cheeger, Finiteness Theorems for Riemannian Manifolds, Amer. J. Math. 92 (1970), 61–74.
  • [CE] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, vol. 9, North-Holland Publishing Co., Amsterdam, 1975.
  • [CFG] J. Cheeger, K. Fukaya, and M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 327–372.
  • [CG1] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986), no. 3, 309–346.
  • [CG2] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990), no. 1, 269–298.
  • [CG3] J. Cheeger and M. Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 115–154.
  • [CR1] J. Cheeger and X. Rong, Collapsed Riemannian manifolds with bounded diameter and bounded covering geometry, Geom. Funct. Anal. 5 (1995), no. 2, 141–163.
  • [CR2] J. Cheeger and X. Rong, Existence of polarized $F$-structures on collapsed manifolds with bounded curvature and diameter, Geom. Funct. Anal. 6 (1996), no. 3, 411–429.
  • [DSW] X. Dai, Z. Shen, and G. Wei, Negative Ricci curvature and isometry group, Duke Math. J. 76 (1994), no. 1, 59–73.
  • [DWY] X. Dai, G. Wei, and R. Ye, Smoothing Riemannian metrics with Ricci curvature bounds, Manuscripta Math. 90 (1996), no. 1, 49–61.
  • [Fu1] K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987), no. 1, 139–156.
  • [Fu2] K. Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988), no. 1, 1–21.
  • [Fu3] K. Fukaya, Collapsing Riemannian manifolds to ones with lower dimension. II, J. Math. Soc. Japan 41 (1989), no. 2, 333–356.
  • [Fu4] K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, Recent topics in differential and analytic geometry, Adv. Stud. Pure Math., vol. 18, Academic Press, Boston, MA, 1990, pp. 143–238.
  • [GY] L. Z. Gao and S.-T. Yau, The existence of negatively Ricci curved metrics on three-manifolds, Invent. Math. 85 (1986), no. 3, 637–652.
  • [Gr1] M. Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978), no. 2, 231–241.
  • [Gr2] M. Gromov, Manifolds of negative curvature, J. Differential Geom. 13 (1978), no. 2, 223–230.
  • [Gr3] M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982), no. 56, 5–99 (1983).
  • [Gro] K. Grove, Metric and topological measurements of manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 511–519.
  • [GPW]1 K. Grove, P. Petersen, and J.-Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), no. 1, 205–213.
  • [GPW]2 K. Grove, P. Petersen, and J.-Y. Wu, Erratum: “Geometric finiteness theorems via controlled topology”, Invent. Math. 104 (1991), no. 1, 221–222.
  • [Ha] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255–306.
  • [Ko] S. Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 70, Springer-Verlag, New York, 1972.
  • [Lo] J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994), no. 3, 655–683.
  • [Mi] M. Min-Oo, Almost Einstein manifolds of negative Ricci curvature, J. Differential Geom. 32 (1990), no. 2, 457–472.
  • [MR] M. Min-Oo and E. Ruh, $L\sp 2$-curvature pinching, Comment. Math. Helv. 65 (1990), no. 1, 36–51.
  • [Pe] G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, II, preprint.
  • [PWY] P. Petersen, G. Wei, and R. Ye, Controlled geometry via smoothing, preprint, 1995.
  • [Ro1] X. Rong, Bounding homotopy and homology groups by curvature and diameter, Duke Math. J. 78 (1995), no. 2, 427–435.
  • [Ro2] X. Rong, On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. (2) 143 (1996), no. 2, 397–411.
  • [Sh] Z. Shen, Deforming metrics with curvature and injectivity radius bounded from below, Arch. Math. (Basel) 62 (1994), no. 4, 354–367.
  • [Shi] W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301.