  This article deals with trace operators on anisotropic Lizorkin--Triebel
spaces with mixed norms over cylindrical domains with smooth boundary. As a
preparation we include a rather self-contained exposition of Lizorkin--Triebel
spaces on manifolds and extend these results to mixed-norm Lizorkin--Triebel
spaces on cylinders in Euclidean space. In addition Rychkov's universal
extension operator for a half space is shown to be bounded with respect to the
mixed norms, and a support preserving right-inverse of the trace is given
explicitly and proved to be continuous in the scale of mixed-norm
Lizorkin--Triebel spaces. As an application, the heat equation is considered in
these spaces, and the necessary compatibility conditions on the data are
deduced.
