On the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg

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The classical Pfaff-Darboux Theorem, which provides local normal forms' for 1-forms on manifolds, has applications in the theory of certain economic models. However, the normal forms needed in these models come with an additional requirement of convexity, which is not provided by the classical proofs of the Pfaff-Darboux Theorem. (The appropriate notion of convexity' is a feature of the economic model. In the simplest case, when the economic model is formulated in a domain in n-space, convexity has its usual meaning. In 2002, Ekeland and Nirenberg were able to characterize necessary and sufficient conditions for a given 1-form to admit a convex local normal form (and to show that some earlier attempts at this characterization had been unsuccessful). In this article, after providing some necessary background, I prove a strengthened and generalized convex Pfaff-Darboux Theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsion-free affine connection on the underlying manifold. (The main result in Ekeland and Nirenberg's paper concerns the case in which the affine connection is flat.)







Robert Bryant

Phillip Griffiths Professor of Mathematics

My research concerns problems in the geometric theory of partial differential equations.  More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.

Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems.  Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of exterior differential systems, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.

I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.

I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.

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