  We construct a decomposition of the identity operator on a Riemannian
manifold $M$ as a sum of smooth orthogonal projections subordinate to an open
cover of $M$. This extends a decomposition of the real line by smooth
orthogonal projection due to Coifman, Meyer and Auscher, Weiss, Wickerhauser,
and a similar decomposition when $M$ is the sphere by the first two authors.
