G2流形上的幾何性質

Abstract

本文主要在研究 G2 流形上的幾何性質，以及與其相關的主題。本文主要分為三個部分，第一部分給出了有關 G2 流形基本性質的定義與重新證明，舉例來說如果一個七維流形是一個 G2 流形的話，那它存在一個愛因斯坦點積，而且是里奇平坦流形。第二部分整理了目前主要在 G2 流形上的體積函數，例如 Hitchin 的體積函數，並分析了其在極值點附近是否有好的性質。第三部分給出了一種在某些主叢上造出愛因斯坦點積的方法，準確來說，我們在上面造出了一個 co-closed G2 結構並滿足了幾乎平行的性質，所以推得在主叢上有一個愛因斯坦點積是由 co-closed G2 結構給出。

In this master thesis, we study the G2 geometry and some relevant topics. There are three main sections in this master thesis, in the first part, we state the definitions and reprove some general facts of G2 geometry, for example, if a 7-dimensional manifold is a G2 manifold, then there exists an Einstein metric on it, moreover, the metric is Ricci flat. In the second part, we summarize some volume functional on G2 manifold in date, for example, the Hitchin's volume functional, and we analyze the stability at the critical points. In the third section, we construct an Einstein metric on certain principal bundle, technically, we give a construction of co-closed G2-structure satisfies the nearly parallel condition, hence the principal bundle contains an Einstein metric which is induced by the co-closed G2-structure.