  We consider second-order linear parabolic operators in non-divergence form
that are intrinsically defined on Riemannian manifolds. In the elliptic case,
Cabr\'e proved a global Krylov-Safonov Harnack inequality under the assumption
that the sectional curvature of the underlying manifold is nonnegative. Later,
Kim improved Cabr\'e's result by replacing the curvature condition by a certain
condition on the distance function. Assuming essentially the same condition
introduced by Kim, we establish Krylov-Safonov Harnack inequality for
nonnegative solutions of the non-divergent parabolic equation. This, in
particular, gives a new proof for Li-Yau Harnack inequality for positive
solutions to the heat equation in a manifold with nonnegative Ricci curvature.
