Geometry and physics of pseudodifferential operators on manifolds

A review is made of the basic tools used in mathematics to define acalculus for pseudodifferential operators on Riemannian manifolds endowed with aconnection: existence theorem for the function that generalizes the phase; analogueof Taylor’s theorem; torsion and curvature terms in the symbolic calculus; the twokinds of derivative acting on smooth sections of the cotangent bundle of the Riemannianmanifold; the concept of symbol as an equivalence class. Physical motivationsand applications are then outlined, with emphasis on Green functions of quantumfield theory and Parker’s evaluation of Hawking radiation.