A review is made of the basic tools used in mathematics to define a calculus
for pseudodifferential operators on Riemannian manifolds endowed with a
connection: esistence theorem for the function that generalizes the phase;
analogue of Taylor's theorem; torsion and curvature terms in the symbolic
calculus; the two kinds of derivative acting on smooth sections of the
cotangent bundle of the Riemannian manifold; the concept of symbol as an
equivalence class. Physical motivations and applications are then outlined,
with emphasis on Green functions of quantum field theory and Parker's
evaluation of Hawking radiation.Comment: 14 pages, paper in honour of Gaetano Vilas
