Higher-Order Curvature and Local Solvability of D(theta).
Let E and F be vector bundles and D:E --> F an operator of order k. We associate a sequence of invariants ((l))(D), l >/= 0 with D which generalize the concept of curvature in a natural way. In the case where D = D(theta) is the differential operator of a connection theta on a vector bundle E, ((1))(D(theta)) is the classical curvature. Furthermore, we find an interesting geometric interpretation for ((2))(D(theta)). Finally, given regularity assumptions, we find, with the aid of these invariants, necessary and sufficient conditions for local solvability of D(theta).
Knapp
R E
RE
Institute for Advanced Study, Princeton, New Jersey 08540.