- #1

victorvmotti

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- TL;DR Summary
- We want to "lift" the pulled-back connection one-form on the base manifold to the total space, but how?!

I am following [this YouTube lecture by Schuller][1] where he finds the appropriate formalism for the quantum mechanics in the physical curved space.

Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold.

He says to the end of the lecture that later on we will see how we can find the connection one-form on the total space, the principal bundle, i.e., the frame bundle, level.

This is not shown, however, in the next video.

I wonder how can we calculate the connection one-form on the total space when we only have information about the pulled back of this form on the base manifold through a section map?

If we have a tangent vector on the base manifold clearly we can push it forward via the section map. But how about when we have defined a form, as we cannot push it forward from the base manifold to the total space via the section map.Is the following a good way to illustrate in the case of a flat base space how to derive the connection one-form on the total space from the given pulled-back connection one-form on the base manifold?

Consider the trivial bundle ##\mathbb{R}^2 \times G \to \mathbb{R}^2##, where ##G## is a Lie group, and let ##E## denote the principal G-bundle with fiber ##G##. We can think of ##E## as the bundle whose fiber over each point in ##\mathbb{R}^2## is just ##G##.

Suppose we have a connection one-form ##\widetilde \omega## on the base manifold ##\mathbb{R}^2## that takes values in the Lie algebra of ##G##. This connection one-form is given by a one-form on ##\mathbb{R}^2## whose values are elements of the Lie algebra of ##G## at each point.

Now suppose we have a section ##s:\mathbb{R}^2 \to E## of the principal G-bundle ##E##. We can think of ##s## as a map that assigns to each point in ##\mathbb{R}^2## an element of ##G##. We can also think of ##s## as a map that takes a point in ##\mathbb{R}^2## and "lifts" it to a point in ##E## by taking the point in ##\mathbb{R}^2## and mapping it to the point in the fiber over that point determined by ##s##.

To derive the connection one-form on the total space, we want to "lift" the pulled-back connection one-form on the base manifold to ##E##. We can do this as follows:

1/ Given a point ##p \in E##, we can use the section map ##s## to identify the point ##q=s^{-1}(p)## in ##\mathbb{R}^2##.

2/ We can then evaluate the pulled-back connection one-form ##\widetilde{\omega} = s^* \omega## on ##\mathbb{R}^2## at the point ##q##. Since ##\widetilde{\omega}## takes values in the Lie algebra of ##G##, this evaluation gives us an element of the Lie algebra of ##G##.

By repeating this process for all points in ##E##, we obtain a connection one-form on ##E## that reduces to ##\omega## when restricted to any fiber of ##E##. This connection one-form satisfies all the conditions required of a connection and can be used to define parallel transport and curvature on ##E##.

[1]:

Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold.

He says to the end of the lecture that later on we will see how we can find the connection one-form on the total space, the principal bundle, i.e., the frame bundle, level.

This is not shown, however, in the next video.

I wonder how can we calculate the connection one-form on the total space when we only have information about the pulled back of this form on the base manifold through a section map?

If we have a tangent vector on the base manifold clearly we can push it forward via the section map. But how about when we have defined a form, as we cannot push it forward from the base manifold to the total space via the section map.Is the following a good way to illustrate in the case of a flat base space how to derive the connection one-form on the total space from the given pulled-back connection one-form on the base manifold?

Consider the trivial bundle ##\mathbb{R}^2 \times G \to \mathbb{R}^2##, where ##G## is a Lie group, and let ##E## denote the principal G-bundle with fiber ##G##. We can think of ##E## as the bundle whose fiber over each point in ##\mathbb{R}^2## is just ##G##.

Suppose we have a connection one-form ##\widetilde \omega## on the base manifold ##\mathbb{R}^2## that takes values in the Lie algebra of ##G##. This connection one-form is given by a one-form on ##\mathbb{R}^2## whose values are elements of the Lie algebra of ##G## at each point.

Now suppose we have a section ##s:\mathbb{R}^2 \to E## of the principal G-bundle ##E##. We can think of ##s## as a map that assigns to each point in ##\mathbb{R}^2## an element of ##G##. We can also think of ##s## as a map that takes a point in ##\mathbb{R}^2## and "lifts" it to a point in ##E## by taking the point in ##\mathbb{R}^2## and mapping it to the point in the fiber over that point determined by ##s##.

To derive the connection one-form on the total space, we want to "lift" the pulled-back connection one-form on the base manifold to ##E##. We can do this as follows:

1/ Given a point ##p \in E##, we can use the section map ##s## to identify the point ##q=s^{-1}(p)## in ##\mathbb{R}^2##.

2/ We can then evaluate the pulled-back connection one-form ##\widetilde{\omega} = s^* \omega## on ##\mathbb{R}^2## at the point ##q##. Since ##\widetilde{\omega}## takes values in the Lie algebra of ##G##, this evaluation gives us an element of the Lie algebra of ##G##.

**3/ We can then "lift" this element to a Lie algebra-valued one-form on ##E## by extending it trivially in the direction transverse to the fiber. Specifically, for any vector ##v## tangent to ##E## at ##p##, we can define the value of the lifted connection one-form at ##p## in the direction of ##v## to be the element of the Lie algebra of ##G## we obtained in step 2.**By repeating this process for all points in ##E##, we obtain a connection one-form on ##E## that reduces to ##\omega## when restricted to any fiber of ##E##. This connection one-form satisfies all the conditions required of a connection and can be used to define parallel transport and curvature on ##E##.

[1]: