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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. 0. Cordes.

Fredholm theory of partial differential equations on complete Riemannian manifolds