  We obtain Harnack estimates for a class of curvature flows in Riemannian
manifolds of constant non-negative sectional curvature as well as in the
Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack
estimate with a bonus term for mean curvature flow in locally symmetric
Riemannian Einstein manifold of non-negative sectional curvature. Using a
concept of "duality" for strictly convex hypersurfaces, we also obtain a new
type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding
flows in the sphere and in the hyperbolic space.
