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|Title: ||On conformal submersions and manifolds with exceptional structure groups|
|Authors: ||Reynolds, Paul|
|Supervisor(s): ||Figueroa-O'Farrill, Jose|
|Issue Date: ||25-Jun-2012|
|Publisher: ||The University of Edinburgh|
|Abstract: ||This thesis comes in three main parts. In the first of these (comprising chapters 2 - 6), the
basic theory of Riemannian and conformal submersions is described and the relevant geometric
machinery explained. The necessary Clifford algebra is established and applied to understand
the relationship between the spinor bundles of the base, the fibres and the total space of a
submersion. O'Neill-type formulae relating the covariant derivatives of spinor fields on the base
and fibres to the corresponding spinor field on the total space are derived. From these, formulae
for the Dirac operators are obtained and applied to prove results on Dirac morphisms in cases
so far unpublished.
The second part (comprising chapters 7-9) contains the basic theory and known classifications
of G2-structures and Spin+
7 -structures in seven and eight dimensions. Formulae relating the
covariant derivatives of the canonical forms and spinor fields are derived in each case. These
are used to confirm the expected result that the form and spinorial classifications coincide. The
mean curvature vector of associative and Cayley submanifolds of these spaces is calculated in
terms of naturally-occurring tensor fields given by the structures.
The final part of the thesis (comprising chapter 10) is an attempt to unify the first two parts.
A certain `7-complex' quotient is described, which is analogous to the well-known hyper-Kahler
quotient construction. This leads to insight into other possible interesting quotients which are
correspondingly analogous to quaternionic-Kahler quotients, and these are speculated upon
with a view to further research.|
|Sponsor(s): ||Engineering and Physical Sciences Research Council (EPSRC)|
|Keywords: ||Riemannian submersions|
|Appears in Collections:||Mathematics thesis and dissertation collection|
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