  In this paper, we introduce the weighted mixed (sectional, Ricci and scalar)
curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$
equipped with a vector field $X$. We define several functions ($q$th Ricci type
curvatures), which "interpolate" between the weighed sectional and Ricci
curvatures. The novel concepts of the "mixed curvature-dimension" condition and
"synthetic dimension of a distribution" allow us to update the estimate of the
diameter of a compact Riemannian foliation and to prove new splitting theorems
for almost-product manifolds of nonnegative/nonpositive weighted mixed scalar
curvature. In the case of positive (and nonnegative) weighted mixed sectional
curvature we explore the weighted generalization of Toponogov's conjecture on
totally geodesic foliations.
