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Riemannian and Totally Geodesic Foliations

  • Philippe Tondeur
Part of the Universitext book series (UTX)

Abstract

The transversal geometry of a foliation is the geometry infinitesimally modeled by Q, while the tangential geometry is infinitesimally modeled by L. A key fact is the existence of the Bott connection in Q defined by
$$ {\mathop{\nabla }\limits^{^\circ }_{{{X^S}}}} = \pi [X,{Y_S}]\,{\text{for}}\,X \in \Gamma L,\,s \in \Gamma Q $$
(5.1)
where YS ∈ ΓTM is any vectorfield projecting to s under π : TM → Q. It is a partial connection along L (only defined for X ∈ ΓL), but otherwise satisfies the usual connection properties. First we observe that the RHS in (5.1) is independent of the choice of YS. Namely the difference of two such choices is a vector field X′ ∈ ΓL, and [X,X′] ∈ ΓL so that π[X,X′] = 0.

Keywords

Normal Bundle Riemannian Foliation Partial Connection Connection Versus Geodesic Foliation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Philippe Tondeur
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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