Non-Embedding and Non-Extension Results in Special Holonomy

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© Oxford University Press 2010. All rights reserved.In the early analyses of metrics with special holonomy in dimensions 7 and 8, particularly in regards to their existence and generality, heavy use was made of the Cartan-Kähler theorem, essentially because the analyses were reduced to the study of overdetermined PDE systems whose natures were complicated by their diffeomorphism invariance. The Cartan-Kähler theory is well suited for the study of such systems and the local properties of their solutions. However, the Cartan-Kähler theory is not particularly well suited for studies of global problems for two reasons: first, it is an approach to PDE that relies entirely on the local solvability of initial value problems and, second, the Cartan-Kähler theory is only applicable in the real-analytic category. Nevertheless, when there are no other adequate methods for analyzing the local generality of such systems, the Cartan-Kähler theory is a useful tool and it has the effect of focusing attention on the initial value problem as an interesting problem in its own right. This chapter clarifies some of the existence issues involved in applying the initial value problem to the problem of constructing metrics with special holonomy. In particular, it discusses the role of the assumption of real-analyticity and presents examples to show that one cannot generally avoid such assumptions in the initial value formulations of these problems.






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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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