Duke Mathematical Journal

A Bochner theorem and applications

Xiaochun Rong

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Article information

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

Dates
First available in Project Euclid: 19 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077232083

Digital Object Identifier
doi:10.1215/S0012-7094-98-09116-5

Mathematical Reviews number (MathSciNet)
MR1600598

Zentralblatt MATH identifier
0962.53026

Subjects
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

Citation

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.


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References

  • [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.