manuscripta mathematica

, Volume 64, Issue 2, pp 213–226 | Cite as

∑-flat manifolds and Riemannian submersions

  • M. Strake
  • G. Walschap


In this paper, we show that a certain rigidity condition (∑-flatness) for open nonnegatively curved manifoldsM is preserved by Riemannian submersions. The result can be applied to quotients ofM by groups of isometries. ∑-flat metrics are also used to derive a splitting theorem for distance tubes of maximal volume growth.


Maximal Volume Number Theory Algebraic Geometry Topological Group Distance Tube 
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  1. 1.
    J. Cheeger,Some examples of manifolds of nonnegative curvature, J. Differential Geometry8 (1973), 623–628Google Scholar
  2. 2.
    J. Cheeger and D. Ebin, “Comparison theorems in Riemannian geometry,” North-Holland, Amsterdam, 1975Google Scholar
  3. 3.
    J. Cheeger and D. Gromoll,On the structure of complete manifolds of nonnegative curvature, Ann. of Math.96 (1972), 413–443Google Scholar
  4. 4.
    J.-H. Eschenburg,Comparison theorems and hypersurfaces, Manuscripta Math.59 (1987), 295–323Google Scholar
  5. 5.
    J.-H. Eschenburg, V. Schroeder, and M. Strake,Curvature at infinity of open nonnegatively curved manifolds, to appear in J. Differential GeometryGoogle Scholar
  6. 6.
    D. Gromoll and K. Grove,The low dimensional metric foliations of Euclidean spheres, to appear in J. Differential GeometryGoogle Scholar
  7. 7.
    V.B. Marenich,The structure of the curvature tensor of an open manifold of nonnegative curvature, Soviet Math. (Doklady)28 (1983), 753–757Google Scholar
  8. 8.
    B. O'Neill,The fundamental equations of a submersion, Michigan Math. J.13 (1966), 459–469Google Scholar
  9. 9.
    W. Poor, “Differential geometric structures,” McGraw-Hill, 1981Google Scholar
  10. 10.
    M. Strake,A splitting theorem for open nonnegatively curved manifolds, Manuscripta Math.61 (1988)Google Scholar
  11. 11.
    G. Walschap,Nonnegatively curved manifolds with souls of codimension 2, J. Differential Geometry27 (1988), 525–537Google Scholar
  12. 12.
    —,A splitting theorem for 4-dimensional manifolds of nonnegative curvature, Proc. Amer. Math. Soc.104 (1988), 265–268Google Scholar
  13. 13.
    J.W. Yim,Space of souls in a complete open manifold of nonnegative curvature, preprintGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • M. Strake
    • 1
    • 2
  • G. Walschap
    • 1
    • 2
  1. 1.State University of New YorkStony Brook
  2. 2.University of CaliforniaLos Angeles

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