Smooth orthogonal projections on Riemannian manifold

TL;DR

This paper develops a method to decompose the identity operator on a Riemannian manifold into smooth orthogonal projections, extending previous decompositions on the real line and the sphere.

Contribution

It introduces a new decomposition technique for the identity operator on Riemannian manifolds using smooth orthogonal projections, generalizing earlier work on simpler spaces.

Findings

Decomposition applies to general Riemannian manifolds.

Extends previous decompositions from real line and sphere.

Provides a framework for localized analysis on manifolds.

Abstract

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.