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Title On conformal submersions and manifolds with exceptional structure groups
Publication Date
Degree PhD
Degree Level doctoral
University/Publisher 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these, formulae for the Dirac operators are obtained and applied to prove results on Dirac morphisms in cases so far unpublished.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.
Subjects/Keywords 519; Riemannian submersions ; conformal submersions ; Clifford algebra ; spinor bundles ; Dirac operators ; quaternionic-Kahler quotients
Rights Full text available
Country of Publication uk
Record ID handle:1842/6218
Repository ethos
Date Indexed 2020-06-17

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…Conventions xi 1 Introduction 1 2 Riemannian Submersions 2.1 Preliminaries . . . . . . 2.2 O’Neill’s tensors . . . . 2.3 Useful properties . . . . 2.4 Examples…

submersions of Riemannian spin manifolds. Does such a generalisation produce a correspondence of spinorial covariant 3 Chapter 1. Introduction derivatives which eliminates not a one-dimensional piece (as for Killing spinors and the cone) but other…

…conformal submersion of Riemannian spin manifolds. We begin by reviewing the theory of Riemannian submersions. These are those maps which preserve both lengths and angles in the horizontal distribution H . The foundational paper of O’Neill [O’N66]…

…generalisation to conformal submersions, of which the cone projection is a special case. The fundamental tensors T g and Ag are defined in the same way as in the Riemannian case and can be related to the tensor fields T λ −2 g −2 and Aλ g of the g…

…cumbersome they are relevant because they are spinorial analogues of maps that preserve the germ of the Laplacian—harmonic Riemannian submersions. Harmonic maps have been studied extensively and we do no more than present some basic definitions and motivation…

…for the theory. More general than harmonic Riemannian submersions are harmonic morphisms, which are maps that preserve the kernel of the Laplacian and not necessarily the entire spectrum. We discuss these briefly as well, and apply Theorem 1.0.4 to…

…summarised below: • Chapter 2: The necessary definitions and basic properties of Riemannian submersions, including O’Neill’s fundamental tensor fields, as well as some examples. • Chapter 3: Similar definitions and facts about conformal submersions, some…

…Spin7 -manifolds by a ‘Clifford quotient’. • Chapter 11: A look at some questions that have arisen during the creation of this thesis but so far remain unanswered. 9 Chapter 1. Introduction 10 Chapter 2 Riemannian Submersions Riemannian