Math @ Duke

Publications [#320294] of Robert Bryant
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 Bryant, RL, On the convex PfaffDarboux Theorem of Ekeland and Nirenberg
(December 22, 2015) [arXiv:1512.07100]
(last updated on 2018/11/15)
Abstract: The classical PfaffDarboux Theorem, which provides local `normal forms' for
1forms 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 PfaffDarboux 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 nspace, convexity has its usual meaning. In 2002,
Ekeland and Nirenberg were able to characterize necessary and sufficient
conditions for a given 1form 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 PfaffDarboux Theorem, one that covers the
case of a Legendrian foliation in which the notion of convexity is defined in
terms of a torsionfree 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.)


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