Solutions to Exercises of Fiber Bundlefrom Liang’s Textbook: Differential Geometry and General Relativity

Abstract.

There is a well-known textbook in Chinese theoretic physics community called Differential Geometry and General Relativity by Liang and Zhou [1] and in Volumn III of the series, there is a chapter on Fiber Bundle. This chapter is a good introduction to the concept of Fiber Bundle for us as beginners and is a good supplement to the book. Here I provide the solutions to the exercises in the chapter.

1. Definitions and Theorems

Definition 1.1.

Principal fiber bundle consists of a bundle manifold $P$, a base manifold $M$ and a structure Lie group $G$. They satisfy the following conditions:

1. $G$ acts freely on $P$ from the right via the action $R:P\times G\to P$;

2. There exists a smooth projection map $\pi :P\to M$ such that for $\forall p\in P$, the fiber over $\pi (p)$ is given by ${\pi }^{-1}\left[\pi (p)\right]=\{pg\mid g\in G\}$;

3. $\forall x\in M$, there exists an open neighborhood $U$ of $x$ in $M$ and a diffeomorphism $T_U:{\pi }^{-1}\left[U\right]\to U\times G$, $T_U\left(p\right)=(\pi \left(p\right),S_U\left(p\right))$, $\forall p\in {\pi }^{-1}\left[U\right]$, where $S_U:{\pi }^{-1}\left[U\right]\to G$ satisfies $S_U\left(pg\right)=S_U\left(p\right)g$, $\forall g\in G$.

Definition 1.2.

Let $R_p:G\to P$, $R_p\left(g\right):-R_g\left(p\right)$, then we can define a fundamental vector field

The definition of a connection on a principal fiber bundle has three equivalent forms:

Definition 1.3.

A connection on a principal fiber bundle $P(M,G)$ assigns to each point $p\in P$ a horizontal subspace $H_p\subseteq T_{p}P$ such that:

1. $T_{p}P=V_p\oplus H_p$, $\forall p\in P$, where $V_p≔\{X\in T_{p}P\mid {\pi }_{\ast }\left(X\right)=0\}$ is vertical subspace;

2. $R_{g\ast }\left[H_p\right]=H_{pg}$, $\forall p\in P$, $\forall g\in G$;

3. $H_p$ is $C^{\mathrm{\infty }}$ with respect to $p$.

Definition 1.4.

A connection on a principal fiber bundle $P(M,G)$ is a $C^{\mathrm{\infty }}$ $𝔤$-valued 1-form on $P$, denoted by $\tilde{\omega }$, such that:

1. ${\tilde{\omega }}_p\left(A_p^{\ast }\right)=A$, $\forall A\in 𝔤$, $\forall p\in P$;

2. Problem 1

Problem 2.1.

Prove Eq.(I-1-4), namely $S_U:{\pi }^{=1}\left[x\right]\to G$ and $R_{{\breve{p}}_U}:G\to {\pi }^{-1}\left[x\right]$ are a pair of inverse mappings.

Proof.

∎

3. Problem 2

Problem 3.1.

Prove Eq.(I-1-5), namely $R_g\circ R_p=R_{pf}$ for $\forall p\in P$, $\forall g\in G$.

Proof.

∎