Continuity of the Alvarez class under deformations

Details Continuity of the Alvarez class under deformations Continuity of the Alvarez class under deformations

2010-09-06

arxiv.org/abs/1009.1098v2

Mathematics

Differential Geometry

31 pages, 2 figures, to appear in J. Reine. Angew. Math.. Proposition 6 was added to correct a mistake in the paragraph after

Scientific paper

A foliated manifold (M,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). Alvarez Lopez defined a cohomology class of degree 1 called the Alvarez class of (M,F) whose triviality characterizes the minimizability of (M,F), when M is closed and F is Riemannian. In this paper, we show that the family of the Alvarez classes of a smooth family of Riemannian foliations is continuous with respect to the parameter. Since the Alvarez class has algebraic rigidity under certain topological conditions on (M,F) as the author showed in arXiv:0909.1125, we show that the minimizability of Riemannian foliations is invariant under deformation under the same topological conditions.

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