Higher-Order Curvature and Local Solvability of D θ | Zendy

Robert E. Knapp | Zendy

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Higher-Order Curvature and Local Solvability of D _θ

Author(s) -

Robert E. Knapp

Publication year - 1973

Publication title -

proceedings of the national academy of sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 5.011

H-Index - 771

eISSN - 1091-6490

pISSN - 0027-8424

DOI - 10.1073/pnas.70.3.797

Subject(s) - curvature , connection (principal bundle) , mathematics , vector bundle , pure mathematics , operator (biology) , differential operator , order (exchange) , sequence (biology) , interpretation (philosophy) , riemann curvature tensor , combinatorics , mathematical analysis , geometry , computer science , biochemistry , chemistry , genetics , finance , repressor , biology , transcription factor , economics , gene , programming language

LetE andF be vector bundles andD :E̱ →F̱ an operator of orderk . We associate a sequence of invariants H⃝(l )(D ),l ≥ 0 withD which generalize the concept of curvature in a natural way. In the case whereD =D θ is the differential operator of a connection θ on a vector bundleE , H⃝(1) (D θ ) is the classical curvature. Furthermore, we find an interesting geometric interpretation for H⃝(2) (D θ ). Finally, given regularity assumptions, we find, with the aid of these invariants, necessary and sufficient conditions for local solvability ofD θ .

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