  This paper describes results characterizing the range of the time-t heat
operator on various manifolds, including Euclidean spaces, spheres, and
hyperbolic spaces. The guiding principle behind these results is this: The
functions in the range of the heat operator should be, roughly, those functions
having an analytic continuation to the appropriate complexified manifold with
growth in the imaginary directions at most like that of the time-t heat heat
kernel.
