Riemannian and Totally Geodesic Foliations

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


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
$$ {\mathop{\nabla }\limits^{^\circ }_{{{X^S}}}} = \pi [X,{Y_S}]\,{\text{for}}\,X \in \Gamma L,\,s \in \Gamma Q $$
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.


Normal Bundle Riemannian Foliation Partial Connection Connection Versus Geodesic Foliation 
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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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