The Euler class for Riemannian flows

Tenseness of Riemannian flows

“…In [27], the Euler class and the Gysin sequence of Riemannian flows on compact manifolds were obtained by using Domínguez's tenseness theorem. Theorems 1.2 and 1.4 allow us to obtain the Euler class and the Gysin sequence of transversally complete Riemannian flows on possibly non-compact manifolds.…”

“…Thus, by Theorem 1.4, it is essential to construct them in the case where the closure of every leaf of F is compact. In this case, with Theorem 1.2, the construction of [27] of the Euler class and the Gysin sequence can be carried out without any modification. Note that, since the closure of every leaf is compact, any leaf has a good saturated neighborhood described by Carrière in [7,Proposition 3], which is called a Carrière neighborhood in [27].…”

“…Here, recall that a foliated manifold (M, F) is called tense if M admits a metric such that the mean curvature form of F is basic. Theorem 1.1 can be regarded as a generalization of Masa's characterization of tautness, and has many applications in the study of geometrical and cohomological properties of Riemannian foliations (see, for example, [17,18,34,27,28,29]). …”

Deformation of Sasakian metrics

“…Saralegui [41] proved that the basic Euler class of F depends only on the smooth type of the flow F (see Royo Prieto [39] for the generalization of the definition of basic Euler classes for Riemannian flows).…”

Tenseness of Riemannian flows

“…The Euler class and the Gysin sequence. In [RP01], the Euler class and the Gysin sequence of Riemannian flows on compact manifolds were obtained by using Domínguez's tenseness theorem. Theorems 1.2 and 1.4 allow us to obtain the Euler class and the Gysin sequence of transversally complete Riemannian flows on possibly non-compact manifolds.…”