Math @ Duke

Publications [#243391] of Robert Bryant
search www.ams.org.Papers Published
 with Bryant, RL; Griffiths, PA; Yang, D, Characteristics and existence of isometric embeddings,
Duke Mathematical Journal, vol. 50 no. 4
(January, 1983),
pp. 893994, Duke University Press [MR85d:53027], [doi]
(last updated on 2019/08/18)
Author's Comments: A study of the characteristic variety of the isometric
embedding
problem in the determined dimension. We show that,
except for metrics whose
Riemann curvature tensor lies in a a very small set of
normal forms, a
3manifold can be isometrically embedded into real
6space. The method
is to show that the system can be made suitably
hyperbolic so that a version
of the NashMoser theorem can be made to apply.
Deane Yang and Johnathan Goodman have since
generalized the principal
analytic result from hyperbolic systems to systems of
real principal type
and have used this to prove isometric embedding
results for 4manifolds.
In higher dimensions, the characteristic variety tends to
have singularities
that no analytic methods are known to handle (in the
smooth category).


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