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.

Definition (Solder form / tautological 1-form)

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

defined by

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:

1. Semibasic: $\theta$ vanishes on vertical tangent vectors (those in $\ker(d\pi)$).
2. 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

1. Euclidean space. On $M=\mathbb{R}^n$ with the global frame $s(x)=\mathrm{id}_{\mathbb{R}^n}$, one has $s^*\theta = (dx^1,\dots,dx^n)$.
2. 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.
3. 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.