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Higher-Order Curvature and Local Solvability of D(theta)
1Institute for Advanced Study, Princeton, New Jersey 08540.
Summary
This study introduces new invariants for differential operators on vector bundles, generalizing curvature. These invariants provide geometric insights and establish conditions for operator solvability.
Area of Science:
- Differential Geometry
- Functional Analysis
- Operator Theory
Background:
- Vector bundles and differential operators are fundamental in modern geometry and analysis.
- Generalizing curvature and understanding operator solvability are key challenges.
Purpose of the Study:
- To introduce a novel sequence of invariants for differential operators on vector bundles.
- To generalize the concept of curvature using these invariants.
- To investigate the geometric interpretation of these invariants and their application to solvability conditions.
Main Methods:
- Associating a sequence of invariants, denoted ((l))(D), with a differential operator D of order k.
- Analyzing the specific case of a differential operator D(theta) associated with a connection theta.
- Deriving necessary and sufficient conditions for local solvability based on the computed invariants.
Main Results:
- A sequence of invariants ((l))(D) is associated with any differential operator D: E --> F.
- The invariant ((1))(D(theta)) corresponds to the classical curvature of the connection theta.
- A novel geometric interpretation for the invariant ((2))(D(theta)) is presented.
- Necessary and sufficient conditions for the local solvability of D(theta) are established.
Conclusions:
- The introduced invariants offer a powerful generalization of curvature for differential operators.
- These invariants provide significant geometric insights.
- The derived conditions for local solvability are crucial for understanding the behavior of differential operators.
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