  Bounds on the logarithmic derivatives of the heat kernel on a compact
Riemannian manifolds have been long known, and were recently extended, for the
log-gradient and log-Hessian, to general complete Riemannian manifolds. Here,
we further extend these bounds to incomplete Riemannan manifolds under the
least restrictive condition on the distance to infinity available, for
derivatives of all orders. Moreover, we consider not only the usual heat kernel
associated to the Laplace-Beltrami operator, but we also allow the addition of
a conservative vector field. We show that these bounds are sharp in general,
even for compact manifolds, and we discuss the difficulties that arise when the
operator incorporates non-conservative vector fields or when the Riemannian
structure is weakened to a sub-Riemannian structure.
