Hiraku Nozawa

Continuity of the Álvarez class under deformations

Abstract A manifold M with a foliation ℱ is minimizable if there exists a Riemannian metric g on M such that every leaf of ℱ is a minimal submanifold of (M, g). For a closed manifold M with a Riemannian foliation ℱ, Álvarez López [1] defined a cohomology class of degree 1 called the Álvarez class whose triviality characterizes the minimizability of (M, ℱ). In this paper, we show that the family of the Álvarez classes of a smooth family of Riemannian foliations on a closed manifold is continuous with respect to the parameter. The Álvarez class has algebraic rigidity under certain topological conditions on (M, ℱ) as the author showed in [21]. As a corollary of these two results, we show that under the same topological conditions the minimizability of Riemannian foliations is invariant under deformations.

Rigidity of the Álvarez class

Let

Minimizability of developable Riemannian foliations

{(M,\mathcal{F})}

Deformation of Sasakian metrics

be a closed manifold with a Riemannian foliation. The Álvarez class of

Tenseness of Riemannian flows

{(M,\mathcal{F})}

Rigidity of the lvarez class

is a cohomology class of M of degree 1 whose triviality characterizes the minimizability or the geometrically tautness of