Frame Bundle
Definition. For an $n$-manifold $M$ (or any rank-$n$ Vector Bundle), the frame bundle $F(M)$ has fiber over $x$ = the set of all ordered bases (frames) of $T_xM$. $GL(n,\mathbb{R})$ acts freely and transitively on frames by change of basis ⇒ $F(M)$ is a Principal Bundle with group $GL(n)$.
Intuition. The concrete, non-exotic principal bundle — good for building intuition before gauge theory. "No preferred point in the fiber" is vivid here: no basis of a tangent space is God-given.
Reductions = geometry (Sternberg's theme). Extra structure on $M$ = reduction of $F(M)$ to a subgroup — this is Sternberg's G-structure:
| Reduce $GL(n)$ to... | Structure on $M$ |
|---|---|
| $O(n)$ | Riemannian metric (orthonormal frames) |
| $GL(n/2, \mathbb{C})$ | almost complex structure |
| $SO(n)$ | orientation + metric |
Why it's in this vault. It closes the loop between bundle types: every vector bundle's twisting data lives in a principal bundle (its frame bundle), and Associated Bundle rebuilds the vector bundle back. Also: GR fits here — the Levi-Civita connection is a connection on the orthonormal frame bundle, making gravity structurally parallel to gauge theory.
Related: Principal Bundle · Vector Bundle · Associated Bundle · Connection on a Principal Bundle
Sources: Sternberg Ch. VII (G-structures — his central object).