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The Euler class for flows of isometries. (English) Zbl 0651.57018
Differential geometry, Proc. 5th Int. Colloq., Santiago de Compostela/Spain 1984, Res. Notes Math. 131, 220-227 (1985).
[For the entire collection see Zbl 0637.00004.]
A flow of isometries is defined as a 1-dimensional orientable Riemannian foliation $${\mathcal F}$$ on a compact manifold M for which there exists a Riemannian metric g on M and a unit vector field Z tangent to $${\mathcal F}$$ generating a group of isometries $$(\psi_ t)$$, $$t\in {\mathbb{R}}$$. The Euler class of $${\mathcal F}$$ is shown to vanish when (M,$${\mathcal F})$$ is a foliated bundle and to be non-zero when $${\mathcal F}$$ is a contact flow (i.e. when there exists a contact form $$\omega$$ on M such that the unique vector field Y on M defind by $$\omega (Y)=1$$ and $$d\omega (Y,\cdot)=0$$ is tangent to $${\mathcal F})$$.
Reviewer: P.Walczak

##### MSC:
 57R30 Foliations in differential topology; geometric theory 57R20 Characteristic classes and numbers in differential topology 53C12 Foliations (differential geometric aspects) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)