Annals of Global Analysis and Geometry

, Volume 1, Issue 3, pp 51–75

The graph of a foliation

  • H. E. Winkelnkemper
Article

DOI: 10.1007/BF02329732

Cite this article as:
Winkelnkemper, H.E. Ann Glob Anal Geom (1983) 1: 51. doi:10.1007/BF02329732

Abstract

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.

Copyright information

© VEB Deutscher Verlag der Wissenschaften 1983

Authors and Affiliations

  • H. E. Winkelnkemper
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege Park

Personalised recommendations