Solder form

See wikipedia.
Definition
Let $M$ be a smooth manifold, and $G$ a Lie group. Let $E \to M$ be a fiber bundle with structure group $G$ (see G-bundles). Suppose that $G$ acts transitively on the standard fibre $F$ , and that $d i m F = d i m M$ . (By the way, the standard fibre would be a homogeneous space).
A soldering of $E$ to $M$ consist of:

A distinguished section $σ : M \to E$ .
A linear isomorphism of vector bundles

θ : T M \to σ^{*} (V E),

from the tangent bundle of $M$ to the pullback of the vertical bundle of $E$ . This is called the solder form.
$◼$
The intuition behind this notion is $E$ represents putting a copy of a homogeneous space on every point of $M$ , and the soldering tell us a point of contact for every $x$ with the homogeneous space and an identification of the tangent spaces.
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Principal bundles

See Wikipedia, down to "principal bundles" section.

Suppose that $M$ has dimension $n$ and $H$ is a group which acts on $R^{n}$ (i.e., we have a group representation of $H$ ). A solder form on a $H$ -principal bundle $P$ over $M$ is an $R^{n}$ -valued 1-form $θ : T P \to R^{n}$ which is horizontal (in the sense that $θ (V) = 0$ for $V$ a vertical vector) and equivariant

R_{h}^{*} θ = h^{- 1} θ

for $h \in H$ ; so that it induces a bundle homomorphism from TM to the associated bundle $P \times_{H} R^{n}$ . This is furthermore required to be a bundle isomorphism.

The reason for the name is that a solder form solders (or attaches) the abstract principal bundle to the manifold M by identifying an associated bundle with the tangent bundle.

The main example is the canonical solder form of the frame bundle, which sends a tangent vector $X \in T_{p} P$ to the coordinates of $d π_{p} (X) \in T_{π (p)} M$ with respect to the frame $p$ .