The Partial Ricci Flow for Foliations

Abstract

We study the flow of metrics on a foliation (called the Partial Ricci Flow), ∂ _t g = −2 r(g), where r is the partial Ricci curvature; in other words, for a unit vector X orthogonal to the leaf, r(X, X) is the mean value of sectional curvatures over all mixed planes containing X. The flow preserves total umbilicity, total geodesy, and harmonicity of foliations. It is used to examine the question: Which foliations admit a metric with a given property of mixed sectional curvature (e.g., constant)? We prove local existence/uniqueness theorem and deduce the evolution equations (that are leaf-wise parabolic) for the curvature tensor. We discuss the case of (co)dimension-one foliations and show that for the warped product initial metric the solution for the normalized flow converges, as t → ∞, to the metric with $r = \Phi \,\hat{g}$, where $\Phi$ is a leaf-wise constant.