Non-Embedding and Non-Extension Results in Special Holonomy
Abstract
© 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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https://hdl.handle.net/10161/13150Published Version (Please cite this version)
10.1093/acprof:oso/9780199534920.003.0017Collections
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Show full item recordScholars@Duke
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 specia

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