Smooth Orthogonal Projections on Riemannian Manifold

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 on the real line by smooth orthogonal projection due to Coifman and Meyer (C. R. Acad. Sci. Paris, Sér. I Math., 312 (3), 259–261 1991 ) and Auscher, Weiss, Wickerhauser ( 1992 ), and a similar decomposition when M is the sphere by Bownik and Dziedziul (Const. Approx., 41 , 23–48 2015 ).