Frame bundle

Definition

Let $M$ be a smooth manifold of dimension $n$ with tangent bundle $TM$. A frame at $p \in M$ is an ordered basis $e = (e_1, \ldots, e_n)$ of the tangent space $T_pM$. The frame bundle of $M$ is the set $$F(M) = \set{(p, e) : p \in M, \ e \text{ a frame at } p}.$$ It comes with the projection $\pi(p, e) = p$ and the right action of the general linear group $GL(n, \R)$ given by $$(p, e) \cdot g = (p, e g), \qquad (e g)_j = \sum_{i=1}^{n} g_{ij} \, e_i.$$ For a chart $(U, x)$ of $M$ and a frame $e$ at $p \in U$, let $A$ be the matrix with $e_j = \sum_i A_{ij} \, \partial/\partial x^i|_p$. $F(M)$ carries the smooth structure for which each map $(p, e) \mapsto (p, A)$ is a diffeomorphism $\pi^{-1}(U) \to U \times GL(n, \R)$. With this structure $F(M)$ is a principal $GL(n, \R)$-bundle over $M$.