Consider a sphere of radius $\sqrt{n}$ in $n$ dimensions, and consider $\vc{X}$, a random variable uniformly distributed on its surface. Poincaré's Observation states that for large $n$, the distribution of the first $k$ coordinates of $\vc{X}$ is close in total variation distance to the standard normal $N(\vc{0},I_k)$. In this paper we consider a larger family of manifolds, and $\vc{X}$ 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.