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.
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