The weighted mixed curvature of a foliated manifold

Abstract

We introduce the weighted mixed curvature of a foliated (and almost-product)
Riemannian manifold equipped with a vector field. We define several $q$th Ricci type curvatures, which interpolate between the weighed sectional and Ricci curvatures. New concepts of the ``mixed curvature-dimensi\-on condition" and ``synthetic dimension of a distribution" allow us to renew the estimate of the diameter of a compact Riemannian foliation and splitting results for almost-product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. We also
study the Toponogov's type conjecture on dimension of a totally geodesic foliation with positive weighted mixed sectional curvature.