Abstract
In this chapter, up to and including §13.4, manifolds need not be compact, or even complete, but must have no boundary. Starting in §13.5, manifolds are once again assumed compact without boundary, unless otherwise stated.
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We say “known” in quotes because in fact the exceptional holonomy groups are excruciatingly difficult to work with, and geometers are nowhere near a classification.
Following our definition of the Cayley numbers on page 154, and the definition of G 2 as the symmetry group of the Cayley numbers, the reader can easily see the invariant 3-form.
But the story is not finished.
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© 2003 Springer-Verlag Berlin Heidelberg
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Berger, M. (2003). Holonomy Groups and Kähler Manifolds. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18245-7_13
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DOI: https://doi.org/10.1007/978-3-642-18245-7_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65317-2
Online ISBN: 978-3-642-18245-7
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