Full Record

Author | Reynolds, Paul |

Title | On conformal submersions and manifolds with exceptional structure groups |

URL | http://hdl.handle.net/1842/6218 |

Publication Date | 2012 |

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these, formulae for the Dirac operators are obtained and applied to prove results on Dirac morphisms in cases so far unpublished.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

Sample Search Hits | Sample Images

…48
One-dimensional fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 *Dirac* *Operators* and Conformal
6.1 *Dirac* operator formulae . . . .
6.2 One-dimensional fibres . . . . .
6.3 *Dirac* morphisms . . . . . . . .
Submersions…

…choice of ξ
The illegal action of ξ
Clifford multiplication of the vector X and the spinor χ
The mean curvature of the vertical distribution V
The mean curvature of the horizontal distribution H
Various *Dirac* *operators*
The real algebra of quaternions
The…

…the *Dirac* *operators*. This does not require us to concoct new machinery and
is a complicated but straightforward calculation. Again presented here in the simplest of four
cases, we find
Theorem 1.0.4. Let π : M → B be a conformal submersion of…

…and ψ be horizontal spinor fields on M, as
in the notation of Theorem 5.1.5, and consider the spinor field on M constructed from these.
The *Dirac* *operators* of the total space, base and fibres are related by
p
1
ˆ ⊗
ˆ χ) − µV · (ϕ ⊗
ˆ χ)…

…with isomorphisms for the general case.
• Chapter 5: Application of the results of the previous two chapters to a derivation of a
spinorial O’Neill formula for any conformal submersion.
• Chapter 6: Calculation of the *Dirac* *operators* and application to…

…formula has been published already in [LS09], but only for the case dim B is even. Its
use there is to characterise so-called *Dirac* morphisms, which are maps that preserve the germ
of the *Dirac* operator. Whilst their definition for dim F > 1 is…

…prove the more difficult analogue of the characterisation of *Dirac*
morphisms proved in [LS09].
This brings to an end the first part of the thesis, although it is hoped many more
applications can be found for the formulae found therein. We now…

…the characterisation of
*Dirac* morphisms.
8
• Chapter 7: Introduction to the background theory of G2 -structures and description of the
form and spinor classifications, as well as a proof of their agreement.
• Chapter 8: Analogous results for Spin7…