Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

On complete manifolds with nonnegative Ricci curvature


Authors: Uwe Abresch and Detlef Gromoll
Journal: J. Amer. Math. Soc. 3 (1990), 355-374
MSC: Primary 53C21
DOI: https://doi.org/10.1090/S0894-0347-1990-1030656-6
MathSciNet review: 1030656
Full-text PDF

References | Similar Articles | Additional Information

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

  • [A] U. Abresch, Lower curvature bounds, Toponogov's Theorem, and bounded topology, I, II, Ann. Sci. Ecole Norm. Sup. (4) 18 (1985), 651-670 and 20 (1987), 475-502. MR 839689 (87j:53058)
  • [An] M. T. Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology (to appear). MR 1046624 (91b:53041)
  • [AW] S. Aloff and N. R. Wallach, An infinite family of distinct $ 7$-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93-97. MR 0370624 (51:6851)
  • [BB] L. Bérard Bergery, Quelques exemples des variétés riemanniennes complètes non compactes à courbure de Ricci positive, C. R. Acad. Sci. Paris Sér. I 302 (1986), 159-161. MR 832061 (87g:53058)
  • [BC] R. Bishop and R. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. MR 0169148 (29:6401)
  • [CE] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, New York, 1975. MR 0458335 (56:16538)
  • [CG1] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413-443. MR 0309010 (46:8121)
  • [CG2] -, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971), 119-128. MR 0303460 (46:2597)
  • [CGT] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15-53. MR 658471 (84b:58109)
  • [EH] J. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Global Anal. Geom. 2 (1984), 141-151. MR 777905 (86h:53042)
  • [Ga1] G. Galloway, A generalization of the Cheeger-Gromoll Splitting Theorem, Arch. Math. 47 (1986), 372-375. MR 866527 (88a:53031)
  • [GM1] D. Gromoll and W. T. Meyer, On complete manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75-90.
  • [GM2] -, Examples of complete manifolds with positive Ricci curvature, J. Differential Geom. 21 (1985), 195-211. MR 816669 (87b:53068)
  • [G] M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), 179-195. MR 630949 (82k:53062)
  • [GLP] M. Gromov, J. Lafontaine, and P. Pansu, Structures métriques pour les variétés Riemanniennes, Nathan, Paris, 1982.
  • [Gr] K. Grove, Metric differential geometry (Nordic Summer School 1985), Lecture Notes in Math., vol. 1263, Springer-Verlag, Berlin and New York, 1987. MR 905882 (88i:53075)
  • [GS] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), 201-211. MR 0500705 (58:18268)
  • [L] H. B. Lawson, The unknottedness of minimal embeddings, Invent. Math. 11 (1970), 183-187. MR 0287447 (44:4651)
  • [M] J. Milnor, A note on curvature and fundamental group, J. Differential Geom. 2 (1968), 1-7. MR 0232311 (38:636)
  • [MS] V. D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces (Appendix: Isoperimetric inequalities in Riemannian manifolds, by M. Gromov), Lecture Notes in Math., vol. 1200, Springer-Verlag, Berlin and New York, 1986. MR 856576 (87m:46038)
  • [MSY] W. Meeks III, L. Simon, and S. T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. 116 (1982), 621-659. MR 678484 (84f:53053)
  • [ShY] J. P. Sha and D. G. Yang, Examples of manifolds of positive Ricci-curvature, J. Differential Geom. 29 (1989), 95-103. MR 978078 (90c:53110)
  • [ShY1] -, Positive Ricci curvature on the connected sums of $ {S^n} \times {S^m}$, J. Differential Geom. (to appear).
  • [SY] R. Schoen and S. T. Yau, Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with nonnegative scalar curvature, Ann. of Math. 110 (1979), 127-142. MR 541332 (81k:58029)
  • [Y1] S. T. Yau, Seminar on differential geometry, Princeton Univ. Press, Princeton, NJ, 1982. MR 645728 (83a:53002)
  • [Y2] -, Some function theoretic properties of complete Riemannian manifolds, and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659-670. MR 0417452 (54:5502)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 53C21

Retrieve articles in all journals with MSC: 53C21


Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1990-1030656-6
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society