  In this paper we establish optimal solvability results, that is, maximal
regularity theorems, for the Cauchy problem for linear parabolic differential
equations of arbitrary order acting on sections of tensor bundles over
boundaryless complete Riemannian manifolds with bounded geometry. We employ an
anisotropic extension of the Fourier multiplier theorem for arbitrary Besov
spaces introduced in earlier by the author. This allows for a unified treatment
of Sobolev-Slobodeckii and little H\"older spaces. In the flat case we recover
classical results for Petrowskii-parabolic Cauchy problems.
