  Consider a sphere of radius root(n) in n dimensions, and consider X, a random
variable uniformly distributed on its surface. Poincare's Observation states
that for large n, the distribution of the first k coordinates of X is close in
total variation distance to the standard normal N(0,I_k). In this paper, we
consider a larger family of manifolds, and X taking a more general distribution
on the surfaces. We establish a bound in the stronger Kullback--Leibler sense
of relative entropy, and discuss its sharpness, providing a necessary condition
for convergence in this sense. We show how our results imply the equivalence of
ensembles for a wider class of test functions than is standard. We also deduce
results of de Finetti type, concerning a generalisation of the idea of
orthogonal invariance.
