Abstract |
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8
dimensions. Riemannian manifolds with
these holonomy groups are Ricci-flat. This is a survey paper on
exceptional holonomy, in two parts.
Part I introduces
the exceptional holonomy groups, and explains constructions for compact
7-
and 8-manifolds with holonomy G2 and
Spin(7). The simplest such constructions work by using techniques from
complex geometry and Calabi--Yau analysis to
resolve the singularities of a torus orbifold T7/ Γ or
T8/ Γ,
for Γ a finite group preserving a
flat G2 or Spin(7)-structure on T7 or T8. There are also more
complicated constructions which begin with a
Calabi–Yau manifold or orbifold.
Part II discusses the calibrated
submanifolds of G2 and Spin(7)-manifolds: associative
3-folds and coassociative 4-folds for G2, and Cayley 4-folds for
Spin(7).
We explain the general theory, following
Harvey and Lawson, and the known examples. Finally we describe the
deformation theory of compact calibrated
submanifolds, following McLean.
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