Higher-Order Curvature and Local Solvability of D_θ

Abstract

Let E and F be vector bundles and D:E̱ → F̱ 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_θ is the differential operator of a connection θ on a vector bundle E, Ⓗ⁽¹⁾(D_θ) is the classical curvature. Furthermore, we find an interesting geometric interpretation for Ⓗ⁽²⁾(D_θ). Finally, given regularity assumptions, we find, with the aid of these invariants, necessary and sufficient conditions for local solvability of D_θ.

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