Rigidity and Splitting of Foliations

Abstract

The known rigidity and splitting results for foliations with nonnegative curvature in mixed directions are not as systematic as those in the theory of Riemannian manifolds of nonnegative curvature; results for foliations usually contain strong additional assumptions, for example, the normal distribution is integrable or totally geodesic, the (co)dimension is equal to one, the curvature is constant, or the result is for a specific class of foliations such as (UF, UF) or (GF, F). In this chapter we begin discussing such results and continue in Chapters VI & VII, which are devoted to a confirmation of the following hypothesis: If the dimension of a foliation is “large” (for example, dim L ≥ ρ(codim L) or dim L > 2 for the Kählerian case) and the mixed sectional curvature is nonnegative, then the situation is locally extremal and we have a metric decomposition of the manifold.