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1. Reynolds, Paul. On conformal submersions and manifolds with exceptional structure groups.

Degree: PhD, 2012, University of Edinburgh

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these, formulae for the Dirac operators are obtained and applied to prove results on Dirac morphisms in cases so far unpublished.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.

Subjects/Keywords: 519; Riemannian submersions; conformal submersions; Clifford algebra; spinor bundles; Dirac operators; quaternionic-Kahler quotients

…115 10 Quotient Constructions 10.1 Hyper-K¨ ahler quotients 10.2 A 7-complex quotient . 10.3… …Clifford structures . . . 10.4 Clifford quotients… …fundamental 4-form on a quaternionic-K¨ ahler manifold or the Clifford 4-form of a Clifford… …structure The covariant derivative induced on E The curvature tensor of ∇E The quaternionic-K… …complex spinors here, to allow for generality in the dimensions. The real and quaternionic cases… 

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APA (6th Edition):

Reynolds, P. (2012). On conformal submersions and manifolds with exceptional structure groups. (Doctoral Dissertation). University of Edinburgh. Retrieved from http://hdl.handle.net/1842/6218

Chicago Manual of Style (16th Edition):

Reynolds, Paul. “On conformal submersions and manifolds with exceptional structure groups.” 2012. Doctoral Dissertation, University of Edinburgh. Accessed December 01, 2020. http://hdl.handle.net/1842/6218.

MLA Handbook (7th Edition):

Reynolds, Paul. “On conformal submersions and manifolds with exceptional structure groups.” 2012. Web. 01 Dec 2020.

Vancouver:

Reynolds P. On conformal submersions and manifolds with exceptional structure groups. [Internet] [Doctoral dissertation]. University of Edinburgh; 2012. [cited 2020 Dec 01]. Available from: http://hdl.handle.net/1842/6218.

Council of Science Editors:

Reynolds P. On conformal submersions and manifolds with exceptional structure groups. [Doctoral Dissertation]. University of Edinburgh; 2012. Available from: http://hdl.handle.net/1842/6218

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