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The graph of a foliation

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.

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Partially supported by NSF Grant MCS77-02721.

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Winkelnkemper, H.E. The graph of a foliation. Ann Glob Anal Geom 1, 51–75 (1983). https://doi.org/10.1007/BF02329732

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Keywords

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