We adapt Topping’s
ℒ-optimal transportation theory for Ricci flow to a more general situation, in which a
complete manifold (M,gij(t)) evolves by ∂tgij= −2Sij, where Sij is a symmetric
2-tensor field on M. We extend some recent results of Topping, Lott and Brendle,
generalize the monotonicity of the 𝒲-entropy of List (and hence also of Perelman),
and recover the monotonicity of the reduced volume of Müller (and hence also of
Perelman).