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Metrics with Exceptional Holonomy

Robert L. Bryant
Annals of Mathematics
Second Series, Vol. 126, No. 3 (Nov., 1987), pp. 525-576
Published by: Annals of Mathematics
DOI: 10.2307/1971360
Stable URL: http://www.jstor.org/stable/1971360
Page Count: 52
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Metrics with Exceptional Holonomy
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Abstract

It is proved that there exist metrics with holonomy G2 and Spin(7), thus settling the remaining cases in Berger's list of possible holonomy groups. We first reformulate the "holonomy H" condition as a set of differential equations for an associated H-structure on a given manifold. We collect the needed algebraic facts about G2 and Spin(7). We then apply the machinery of over-determined partial differential equations (in the form of the Cartan-Kahler theorem) to prove the existence of solutions whose holonomy is G2 or Spin(7). We also provide explicit examples and some information about the "generality" of the space of such metrics.

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