Riemannian metrics and Laplacians for generalized smooth distributions

Abstract

In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation C * -algebra. To this end, we give the construction of a parametrix.1991 Mathematics Subject Classification. 58J60, 53C17, 46L08, 58B34.

Coordinates for non-integrable Lie algebroids

“…Orthonormality here makes sense up to the choice of a smooth fiberwise inner product on the original Lie algebroid A. Indeed, it was shown in [AK19] that such a choice gives rise to a family of inner products p¨, ¨qy on F y for every y P V , which is smooth in a well defined way.…”