Smolyanov surface measures and solutions of Schroedinger and heat type equations on a Riemannian manifold

In the paper we give solutions of Schroedinger and heat type equations on a Riemannian manifold in the form of functional integrals over Smolyanov surface measures on the set of trajectories in a manifold and over the Wiener measure, generated by Brownian motion in the manifold. These integrals are in fact limits of finite-dimensional integrals (over Cartesian products of some copies of the manifold) of elementary functions containing coefficients of equations, initial conditions and geometric characteristics of the manifold. In the proof, a substantial role is played by Smolyanov-Weizsaecker-Wittich asymptotic estimates for Gaussian integrals over a manifold, by the Chernoff theorem and by the method of transition from the Schroedinger to the heat equation going back to Doss.