  We study the reduction of the heat equation in Riemannian spaces which admit
a gradient Killing vector, a gradient homothetic vector and in Petrov Type
D,N,II and Type III space-times. In each reduction we identify the source of
the Type II hidden symmetries. More specifically we find that a) If we reduce
the heat equation by the symmetries generated by the gradient KV the reduced
equation is a linear heat equation in the nondecomposable space. b) If we
reduce the heat equation via the symmetries generated by the gradient HV the
reduced equation is a Laplace equation for an appropriate metric. In this case
the Type II hidden symmetries are generated from the proper CKVs. c) In the
Petrov spacetimes the reduction of the heat equation by the symmetry generated
from the nongradient HV gives PDEs which inherit the Lie symmetries hence no
Type II hidden symmetries appear. We apply the general results to cases in
which the initial metric is specified. We consider the case that the
irreducible part of the decomposed space is a space of constant nonvanishing
curvature and the case of the spatially flat Friedmann-Robertson-Walker space
time used in Cosmology. In each case we give explicitly the Type II hidden
symmetries provided they exist.
