This paper studies necessary
and sufficient conditions for differential operators to be Fredholm on the Sobolev
spaces of a complete (not necessarily compact) Riemannian manifold Ω. The
conditions are formulated algebraically in terms of the nonvanishing of the operator’s
principal symbol on Ω (ellipticity) and its “total symbol” at infinity of Ω. The
operators considered arise by taking sums of products of vector fields, all of whose
covariant derivatives vanish at infinity; and the study involves C∗-algebra
techniques. The required technical restrictions on the curvature and topology of Ω
near infinity are much weaker than those in earlier joint work with H. O.
Cordes.
