Fundamental Solution Operator for the Cauchy Problem on a Manifold

In this chapter we construct the fundamental solution operator \(U(t) = e^{-t\,P}\) of the Cauchy problem (CP) for the heat operator on an n-dimensional, compact smooth Riemannian manifold \((M, g)\) without boundary. In the study of the Cauchy problem (CP), upon using local coordinate systems flattening out the manifold \((M, g)\) together with a partition of unity we are reduced to the Euclidean space case \({\mathbf {R}}^{n}\) in a standard fashion. In this way we are reduced to the study of the individual terms each of which can be analyzed in local coordinates (Theorems 15.2 and 15.4).