Reference Request for Fibre Bundle Theory from the Smooth Manifold Point of View

I am looking for a book, or a set of notes, which discusses some basic theory of fibre bundles. I am interested more in the geometric aspect (smooth manifolds) rather than topological aspect.

I found Steenrod's Topology of Fiber Bundles which covers the topology part in quite some depth but nothing which discusses the smooth aspect.

Can anybody point me to a nice source?

From personal experience I think there's a terrible gap in the literature in this case. That being said, you might find part of what you're looking for in Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. It advocates the priciple of unifying the different aspects of differential geometry via $G$-bundles and transport information (connection).

I'd like to say something here about the subject at hand since I feel like it's not so easy to find sources explaining this stuff.

In all the books I've seen a connection on a principal bundle is defined as a lie algebra valued form satisfying certain equivariant conditions. This is very confusing for someone whose learning about them in the first time. Here's a simple way of seeing why these conditions are natural.

Every principal $G$-bundle $\pi: P \to M$ naturally gives rise to the following exact sequence of vector bundles over $P$.

$$0 \to VP \to TP \to \pi^*TM \to 0$$

Where $VP = \ker \pi_*$ (called the vertical bundle) and the maps are the obvious ones.

A very general and important thing to remember is that every exact sequence of vector bundles splits. The splitting though is not natural most of the times.

Definition: A connection on $P$ is a choice of splitting of the above sequence.

Recall that from general abstract nonsense a splitting could be desribed equivalently either as a retraction of $VP \to TP$ or as a section of $TP \to \pi^*TM$.

Question: How do lie algebra valued forms enter into this picture?

Well, there's a very simple answer here but it requires a litlle work to actually prove it's correct. To avoid this I'm goning to shamelessly use the following lemma without proving it.

Lemma: There's a cannonical equivariant isomorphism of vector bundles $VP \cong ad(P)$. Where $ad(P)=P\times_{ad}\mathfrak{g}$ is the equivariant bundle on $P$ associated to the adjoint representation of $G$ on its lie algebra $ad:G \to GL(\mathfrak{g})$.

This is a strengthening of the well known statement about the 1-1 correspondence between vertical vector fields on $P$ and elements of the lie algebra (the vertical vector fields are called fundamental vector fields in this case). The proof here is simple enough, the map is the one obtained from differentiating the action of $G$ and all that's left is verifying the equivariance which is just a computation (you might want to come back later and prove this after youv'e had some more experience with this stuff).

Assuming the above lemma we see that giving a retraction for $VP \to TP$ is the same as giving a map $\omega: TP \to ad(P)$ that restricts to the isomorphism from the lemma i.e. $\omega|_{VP} VP \cong ad(P)$ (this condition is often denoted by $\omega(X^*)=X$ where $X^*$ is the fundamental vector field corresponding to $X \in \mathfrak{g}$).

That's it! $\omega \in \Omega^1(P,\mathfrak{g})$ is our connection form. The rest of the requirements on $\omega$ are taken care of by the transformation properties of the vector bundle $ad(P)$.

Where is the horizontal bundle? Well, since $\omega$ projects to the vertical bundle the horizontal bundle is simply $\ker \omega$.

I hope this was helpful to you. Good luck.

• Thank you. Very helpful post. – caffeinemachine Dec 27 '15 at 5:53

You can consult:

1. Husemöller - Fibre Bundles;
2. Koszul - Lectures On Fibre Bundles and Differential Geometry, available at http://www.math.tifr.res.in/~publ/ln/tifr20.pdf;
3. Spivak - A Comprehensive Introduction to Differential Geometry, only volumes 1 and 2.

And if you are sufficiently experienced, you can consult:

1. Kobayashi, Nomizu - Foundations of Differential Geometry, both two volumes;
2. Kolár, Michor, Slovák - Natural Operations in Differential Geometry, available at http://www.emis.de/monographs/KSM/kmsbookh.pdf;
3. Milnor, Stasheef - Characteristic classes, a very easy text but very deep!