Foliations on Riemannian Manifolds pp 132-142 | Cite as

# Flows

Chapter

## Abstract

Let F be a tangentially oriented foliation of dimension one on (M,g is the characteristic form of F. The induced metric 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 ∈ Ω^{1}(M) defined by$$ \chi ({\text{Y}}) = {g_M}({\text{T,Y}})\,{\text{for}}\,{\text{Y}} \in \Gamma {\text{TM}} $$

(10.1)

_{L}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