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Flows

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

Abstract

Let F be a tangentially oriented foliation of dimension one on (M,gM). Such a foliation is called a flow. The leaves of F are the integral curves of a nonsingular vector field X on M. Normalizing length shows that F is also given by a unit vector field T with respect to gM. The dual 1-form χ ∈ Ω1(M) defined by
$$ \chi ({\text{Y}}) = {g_M}({\text{T,Y}})\,{\text{for}}\,{\text{Y}} \in \Gamma {\text{TM}} $$
(10.1)
is the characteristic form of F. The induced metric gL is related to χ by
$$ {g_L}(\lambda {\text{T,}}\lambda {\text{T}}) = {\lambda^2},\,\chi {(}\lambda {\text{T) = }}\lambda $$
(10.2)
.

Keywords

Vector Field Exact Sequence Integral Curve Holonomy Group Linear Flow 
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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