Flows

Abstract

Let F be a tangentially oriented foliation of dimension one on (M,g_M). 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 g_M. 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 g_L is related to χ by

$$ {g_L}(\lambda {\text{T,}}\lambda {\text{T}}) = {\lambda^2},\,\chi {(}\lambda {\text{T) = }}\lambda $$

((10.2))

.