Calibrated foliations and invariant metrics. (Q1862725)

The idea of calibration theory is contained in Huyghens algorithm for a family of light rays in a nonhomogeneous environment. In this paper, the theory of calibration is used for study of invariant foliations on Riemannian manifolds. Problem: Given an action of a compact Lie group \(G\) on a manifold \(X\), let \({\mathcal F}\) be an oriented, \(G\)-invariant, \(K\)-dimensional foliation on \(X\). Can one find a closed \(K\)-form \(\omega\) and a \(G\)-invariant Riemannian metric with respect to which \(\omega\) becomes a calibration to \({\mathcal F}\)? The author proves sufficient conditions for an affirmative answer of this problem, and he considers calibrated foliations on symmetric spaces. Let \(X\) be a compact Riemannian symmetric space, \(G\) a connected group of isometries of \(X\), and \(H\) is a stationary subgroup. A result of this paper is the following theorem: Let \(X= G/H\) a compact Riemannian symmetric space, \({\mathcal F}\) is a totally geodesic foliation, \(G\)-invariant, on \(X\). Then there exists a \(G\)-invariant metric on \(X\) in which the foliation \({\mathcal F}\) can be calibrated. Finally, the author studies the relationship of the existence of invariant metrics with the exponential growth of leaves of a tight foliation. Theorem: Let a compact Lie group \(G\) act on a compact manifold \(X\) and let \({\mathcal F}\) be a tight, \(G\)-invariant foliation of dimension \(K\) on the manifold \(X\) whose \(K\)th homology group (over \(\mathbb{R}\)) is zero. Then there exists a \(G\)-invariant metric on \(X\) in which the growth of each leaf of \({\mathcal F}\) must be exponential.