Solder form on the frame bundle

Let $M$ be an $n$ -dimensional smooth manifold. Its (linear) frame bundle $FM$ is the set of pairs $(x,u)$ where $x\in M$ and $u\colon \mathbb{R}^n\to T_xM$ is a linear isomorphism. The projection $\pi\colon FM\to M$ makes $FM$ a principal bundle with structure group $GL(n,\mathbb{R})$ acting by right composition on frames.

The solder form on $FM$ is the $\mathbb{R}^n$ -valued 1-form

\theta \in \Omega^1(FM;\mathbb{R}^n)

defined by

\theta_u(v) := u^{-1}\bigl(d\pi_u(v)\bigr), \qquad u\in FM,\; v\in T_uFM.

Equivalently: $\theta$ “measures” the base component of a tangent vector to $FM$ and expresses it in the moving frame $u$ .

The solder form satisfies two fundamental properties:

Semibasic: $\theta$ vanishes on vertical tangent vectors (those in $\ker(d\pi)$ ).
Equivariance: for $A\in GL(n,\mathbb{R})$ and the right action $R_A\colon FM\to FM$ , one has $(R_A)^*\theta = A^{-1}\theta.$

A local section $s\colon U\to FM$ (a local frame) pulls back $\theta$ to a coframing on $U$ , i.e. an identification $TU\cong U\times \mathbb{R}^n$ compatible with the tangent bundle.

Examples

Euclidean space. On $M=\mathbb{R}^n$ with the global frame $s(x)=\mathrm{id}_{\mathbb{R}^n}$ , one has $s^*\theta = (dx^1,\dots,dx^n)$ .
Orthonormal frame bundle. If $M$ is Riemannian, restricting $FM$ to orthonormal frames yields a principal $O(n)$ -bundle; the same formula defines the solder form on this subbundle.
Pullback to a moving frame. On any coordinate chart $U\subset M$ , taking $s$ to be the coordinate frame gives $s^*\theta$ equal to the usual coordinate coframe, showing that $\theta$ globalizes the local “ $dx$ ” data.