特殊和樂群流形及其校準子流形

Abstract

本論文為將介紹帶有特殊和樂群G2與Spin(7)的流形，利用校準以及八元數對此主題進行深入探討，最後討論結合子流形、餘結合子流形、Cayley子流形的形變向量場。

This thesis is a brief survey of manifolds with exceptional holonomy groups G2 and Spin(7). These two holonomy groups come from Berger’s classification [2]. In chapter 2, I introduce some basic properties of the group G2 and Spin(7), most of these results and proofs are from [4], [6], [7]. Chapter 3 is an introduction to the notion of calibration and octonions, and use octonion to discover more insights of the G2 and Spin(7) geometry. The examples of calibrated submanifolds we are going to study are associative, coassociative and Cayley submanifolds. Chapter 4 gives a discussion about the deformation vector fields of these calibrated submanifolds, which is from Mclean’s paper [8].