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

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

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

Personalised recommendations