Skip to main content

The graph of a foliation


Let M be a riemannian manifold with a riemannian foliation F. Among other things we construct a special metric on the graph of the foliation,\(\mathfrak{G}(F)\), (which is complete, when M is complete), and use the relations of Gray [1] and O'Neill [7] and the elementary structural properties of\(\mathfrak{G}(F)\), to find a necessary and sufficient condition that\(\mathfrak{G}(F)\) also have non-positive sectional curvature, when M does.

This condition depends only on the second fundamental form and the holonomy of the leaves.

As a corollary we obtain a generalization of the Cartan-Hadamard Theorem.

This is a preview of subscription content, access via your institution.


  1. [1]

    GRAY, A., Pseudo-Riemannian Almost Product Manifolds and Submersions, Journal of Math. and Mech. 16 (1967), 715–738.

    MATH  Google Scholar 

  2. [2]

    HAEFLIGER, A., Variétés Feuilletées, Ann. Scuola Norm. Sup. Pisa 16 (1962), 367–307.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    HERMANN, R., On the Differential Geometry of Foliations,Ann. of Math. 72 (1960), 445–457.

    MATH  MathSciNet  Google Scholar 

  4. [4]

    HERMANN, R., A sufficient conditions that a mapping or riemannian manifolds be a fiber bundle, Proc. A.M.S. 11 (1960), 236–242.

    MATH  MathSciNet  Google Scholar 

  5. [5]

    MILNOR, J., Morse Theory, Princeton U. Press, 1963.

  6. [6]

    MILNOR, J. and STASHEFF, J., Characteristic Classes, Princeton U. Press, 1974.

  7. [7]

    O'NEILL, B., The Fundamental Equations of a Submersion, Michigan Math. J. 13 (1966), 459–469.

    MATH  MathSciNet  Google Scholar 

  8. [8]

    REINHART, B., Foliated Manifolds with Bundle Like Metrics, Ann. of Math. 69 (1959), 119–132.

    MATH  MathSciNet  Google Scholar 

  9. [9]

    THOM, R., frGénéralisation de la Théorie de Morse aux Variétés Feuilletées, Ann. Inst. Fourier Grenoble 14 (1964), 173–190.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    WEINSTEIN, A., Symplectic Geometry, Bull, A.M.S. (new series) 5 (1981), 1–12.

    MATH  MathSciNet  Google Scholar 

Download references

Author information



Additional information

Partially supported by NSF Grant MCS77-02721.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Winkelnkemper, H.E. The graph of a foliation. Ann Glob Anal Geom 1, 51–75 (1983).

Download citation


  • Structural Property
  • Riemannian Manifold
  • Group Theory
  • Sectional Curvature
  • Fundamental Form