# Annoying principal bundle problem

• Fredrik
In summary: So how can we say that (6.2.14) is true?In summary, the author of this book (Isham) uses a curve to construct a pullback bundle, and he proves that the integral curve of this pullback bundle is also a section.
Fredrik
Staff Emeritus
Gold Member
Consider a principal bundle $\pi:E\rightarrow M$ with structure group G. Let $I=(a-\epsilon,b+\epsilon)$. We can use a curve $\alpha:I\rightarrow M$ to construct a pullback bundle $Pr_1:IE\rightarrow I$. I'm calling the total space of the pullback bundle IE because that reminds me that it's a subspace of $I\times E$. To be more specific, we have

$$IE=\{(t,p)\in I\times E|\alpha(t)=\pi(p)\}$$

This pullback bundle is a principal bundle too, and $Pr_2$ is a principal bundle isomorphism. If $\omega$ is a connection 1-form on the original principal bundle, then $Pr_2^*\omega$ is a connection 1-form on the pullback bundle. (I was able to prove that, so that's not what I'm going to ask about).Now let D be the derivative operator that takes a real-valued function on I to its derivative, i.e. Df=f' and Dtf=f'(t). This D is a vector field on I. The "horizontal lift" of this vector field is defined as the unique horizontal vector field $D^H$ that satisfies the condition $\pi_*D^H=D$. "Horizontal" means that it's taken to 0 by the connection 1-form, so we have $(Pr_2^*\omega)_{(t,p)}D^H_{(t,p)}=0$.I understand all of the above, but there's another detail that looks like it should be easy to prove, and yet I can't figure it out. Define $\gamma$ to be the integral curve of $D^H$ trough (a,p) (where a is the real number mentioned in the definition of I, and p is some arbitrary point in E). This $\gamma$ is a curve in IE, so we can write $\gamma(t)=(\gamma_I(t),\gamma_E(t))$, but why is $\gamma_I(t)=t$?I'm stuck at a "clearly we have" statement in the book I'm reading, and that statement is only true if $\gamma_I(t)=t$, so apparently the author of this book (Isham) felt that this is so trivial that he didn't even need to include a "clearly we have" statement about that. I'm hoping that someone can tell me why the integral curve $\gamma$ must also be a section.This stuff is on page 264 of "Modern differential geometry for physicists" by Chris Isham, but for some reason it isn't possible to view page 264 either at Google books or at Amazon. If you happen to own a copy of the book, you might get confused by the fact that my notation is different from his.

Last edited:
I think I got it. What I want to prove is that $Pr_1\circ\gamma$ is the identity map on I. First note that

$$(Pr_1\circ\gamma)_*D_t=Pr_1_*\gamma_*D_t=Pr_1_*\dot\gamma_{\gamma(t)}=Pr_1_*D^H_{\gamma(t)}=(Pr_1_*D^H)_{Pr_1(\gamma(t))}=D_{Pr_1\circ\gamma(t)}$$

So if we write $\sigma=Pr_1\circ\gamma$, we have

$$\sigma_*D_t=D_{\sigma(t)}$$

but $\sigma$ is a function from I into I (and I is an open interval in the field of real numbers), so the chain rule tells us that we also have

$$\sigma_*D_t f=D_t(f\circ\sigma)=(f\circ\sigma)'(t)=f'(\sigma(t))\sigma'(t)=D_{\sigma(t)}f\sigma'(t)$$

i.e.

$$\sigma_*D_t=\sigma'(t)D_{\sigma(t)}$$

and a comparison with the first result shows that $\sigma'(t)=1$ for all t, which means that $\sigma=id_I+C$, where C is a constant. The constant can be determined from the condition $\gamma(0)=(a,p)$, which implies that

$$Pr_1\circ\gamma(0)=Pr_1(a,p)=a[/itex] So we have C=a, which isn't what I expected. Here's what Isham does: He defines $\alpha^H=Pr_2\circ\gamma$ and then says that "clearly we have" $\pi(\alpha^H(t))=\alpha(t)$, but if I'm right, this isn't just "not clear", it's wrong. We have [tex]\gamma(t)=(Pr_1(\gamma(t)),Pr_2(\gamma(t)))=(\sigma(t),\alpha^H(t))$$

and therefore

$$\pi(\alpha^H(t))=\pi\circ Pr_2\circ\gamma(t)=\pi(\alpha^H(t))=\alpha(\sigma(t))=\alpha((id_I+a)(t))=\alpha(t+a)$$

New question:

I ran into some difficulties on page 266 too. Can someone explain (6.2.14) to me? He mentions (6.1.26). There's no preview for that page, but he writes that gauge transformation like this:

$$A_\mu(x)\mapsto\Omega(x)A_\mu(x)\Omega(x)^{-1}+\Omega(x)\partial_\mu\Omega(x)^{-1}$$

Both A and $\Omega$ are matrix-valued functions.

## 1. What is an annoying principal bundle problem?

An annoying principal bundle problem is a mathematical concept that arises in the study of fiber bundles, which are geometric structures used to describe the behavior of objects that vary over a space. It refers to a situation where the principal bundle associated with a particular fiber bundle is not unique, making it difficult to study and analyze.

## 2. How does an annoying principal bundle problem affect scientific research?

An annoying principal bundle problem can significantly impact scientific research, especially in fields such as physics and mathematics, where fiber bundles are commonly used. It can make it challenging to accurately model and analyze complex systems, leading to potential errors or incomplete understanding.

## 3. What causes an annoying principal bundle problem?

An annoying principal bundle problem can arise due to a variety of factors, such as the non-uniqueness of the principal bundle or the choice of coordinate charts used to describe the bundle. It can also be caused by the non-trivial topology of the base space, which can complicate the identification of the principal bundle.

## 4. How do scientists solve an annoying principal bundle problem?

Solving an annoying principal bundle problem requires a deep understanding of the underlying mathematical principles and the specific context in which it arises. Scientists may use techniques such as fiber integration or homotopy theory to simplify the problem and identify the correct principal bundle.

## 5. Are there any real-world applications of the annoying principal bundle problem?

Yes, the concept of an annoying principal bundle problem has real-world applications, particularly in fields such as quantum mechanics and gauge theory, where fiber bundles play a crucial role. By understanding and addressing this problem, scientists can develop more accurate and comprehensive models of complex systems and phenomena.

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