Heat equations on Lie groups, symmetric spaces and Riemannian manifolds

In the present paper we prove several existence and uniqueness theorems for Cauchy problems of heat equations on compact connected Lie groups, Heisenberg groups, noncompact Riemannian manifolds and homogeneous spaces of constant curvature.; For compact connected Lie groups, the Cauchy data are in the category of hyperfunctions, and we can consider that our results are in the most general form in several form for these theorems.; Furthermore, we show that compactness or noncompactness of the underlying manifold is a crucial condition for uniqueness of the solutions of the heat equations. This provides a new insight of uniqueness of solutions of heat equations.; Finally, as applications, we obtain several results in harmonic analysis including Bochner-Godement type theorems and Schwartz type kernel theorems on compact Lie groups. Those results guide us to apply the theory of heat equations to several aspects of harmonic analysis.; For the Heisenberg groups, we have qualitative properties of solutions of the heat equations which generalize some results of Widder for the Lie groups.