  A rather complete investigation of anisotropic Bessel potential, Besov, and
H\"older spaces on cylinders over (possibly) noncompact Riemannian manifolds
with boundary is carried out. The geometry of the underlying manifold near its
'ends' is determined by a singularity function which leads naturally to the
study of weighted function spaces. Besides of the derivation of Sobolev-type
embedding results, sharp trace theorems, point-wise multiplier properties, and
interpolation characterizations particular emphasize is put on spaces
distinguished by boundary conditions. This work is the fundament for the
analysis of time-dependent partial differential equations on singular
manifolds.
