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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

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.)