Why do people care about principal bundles?

Solution 1:

For 1, given any free action of a compact Lie group $G$ on a manifold $M$ (or a free proper action of a noncompact Lie group), the orbit space $M/G$ naturally has the structure of a smooth manifold such that the projection $\pi:M\rightarrow M/G$ is a smooth submersion. (Free means the only group element which fixes at least one point is the identity).

It turns out that $\pi$ is actually a $G$-principal fiber bundle. So, if you care about group actions on manifolds, principal bundles arise naturally.

For 3, one of the main uses of the frame bundle I know of is the following: Suppose a compact Lie group $G$ acts effectively on a Riemannian manifold $M$. (Effective means the only group element which fixes all points is the identity). Since the action is not free, the orbit space $M/G$ isn't a manifold in any kind of natural way (though it is still not so bad as a topological space!).

On the other hand, the action induces an action on the tangent bundle $TM$ (which still might not be free), and induces and action on the frame bundle $FM$. This induced action on the frame bundle is free, so the quotient $FM/G$ is a manifold, so all the tools of differential geometry can be used to study $FM/G$, which in turn can give information about $M$.

Solution 2:

Off the top of my head I can think of a few reasons:

1. For vector bundles there is the useful notion of a connection 1-form. This is a little messy since it is dependent on a frame. For principal bundles however, a connection 1-form is a global, well-defined object.
2. They are useful in defining bundles. For example if you have a principal-$G$ bundle $P$ over $M$ and a representation $V$ of $G$, you get a vector bundle over $M$ and a connection on the principal bundle induces a connection on the vector bundle. These sorts of vector bundles are often easier to work with because their sections are just $G$ invariant functions from $P \to V$. These constructions are vital in things like spin geometry.
3. It is also useful from an algebraic topological point of view. General characteristic classes can be defined for a group $G$ which then give characteristic classes for any vector bundle associated to a principal $G$-bundle.
4. Geometric structures on a vector bundle can be thought of as reductions of the frame bundle. For example a Riemannian metric is a reduction to $O(n)$, a complex structure is a reduction to $GL(n,\mathbb C)$, etc.

EDIT:

Here's a nice concrete example combining my point 2 with Jason's answer. Let $G$ be a compact semisimple Lie group and $T$ a maximal torus. An irreducible representation of $G$ is determined by it's highest weight $\mu$ which determines a map $e^\mu: T \to \mathbb C^\times$, i.e. a one-dimensional complex representation of $T$. By Jason's point $G \to G/T$ is a principal $T$-bundle so by my point 2, you can form the vector bundle associated to $e^\mu$. The group $G$ acts on sections of this vector bundle and the Borel-Weil theorem says that the space of holomorphic sections is the representation of $G$ with highest weight $\mu$. So in using the principal bundle $G \to G/T$, we can geometrically construct all of the representations of $G$ explicitly.

Solution 3:

In string theory, principal bundles can arise quite naturally when considering the compactification of extra dimensions.

T-Duality in string theory is a duality which relates string theory on one spacetime to string theory on another spacetime. The simplest example of T-duality says that bosonic string theory on $\mathbb{R}^{25} \times S_{R}^1$ is equivalent (in terms of the physical observables of the theory) to bosonic string theory on $\mathbb{R}^{25} \times S_{1/R}^1$.

(The subscript on the $S^1$ refers to the radius of the compactified dimension.)

A natural generalisation of T-duality considers a spacetime with a free circle action whose orbit space is the uncompactified spacetime manifold $M$. Locally, spacetime is a product $M \times S^1$, but not globally.

One is led to consider a principal $S^1$ bundle $\pi : E \to M$, where $M$ is the uncompactified part of your spacetime, and $E$ is your total spacetime. T-duality then refers to the construction of another principal bundle $\hat{\pi}: \hat{E} \to M$ on which you have an equivalent string theory.