Returning to our motivating example, the **tangent bundle** on a manifold \({M^{n}}\), denoted \({TM}\), is a smooth vector bundle \({(E,M^{n},\mathbb{R}^{n})}\) with a (possibly reducible) structure group \({GL(n,\mathbb{R})}\) that acts as an inverse change of local frame across trivializing neighborhoods. These trivializing neighborhoods can be obtained from an atlas on \({M}\), with fiber homeomorphisms \({f_{i}\colon T_{x}M\rightarrow\mathbb{R}^{n}}\) defined by components in the coordinate frame \({e_{i\mu}=\partial/\partial x_{i}^{\mu}}\), so that the transition functions are Jacobian matrices

\(\displaystyle \begin{aligned}v_{i}^{\mu} & =(g_{ij})^{\mu}{}_{\lambda}v_{j}^{\lambda}\\ & =\frac{\partial x_{i}^{\mu}}{\partial x_{j}^{\lambda}}v_{j}^{\lambda} \end{aligned} \)

associated with the transformation of vector components. \({M}\) is orientable iff these Jacobians all have positive determinant, i.e. iff the structure group is reducible to \({GL(n,\mathbb{R})^{e}}\) (the definition of \({TM}\) being orientable). A section of the tangent bundle is a vector field on \({M}\). A change of coordinates within each coordinate patch then generates a change of frame

\(\displaystyle \frac{\partial}{\partial x_{i}^{\prime\mu}}=\frac{\partial x_{i}^{\lambda}}{\partial x_{i}^{\prime\mu}}\frac{\partial}{\partial x_{i}^{\lambda}}, \)

which is equivalent to new local trivializations where

\(\displaystyle v_{i}^{\prime\mu}=\frac{\partial x_{i}^{\prime\mu}}{\partial x_{i}^{\lambda}}v_{i}^{\lambda}, \)

giving us new transition functions

\(\displaystyle \frac{\partial x_{i}^{\prime\mu}}{\partial x_{j}^{\prime\lambda}}=\frac{\partial x_{i}^{\prime\mu}}{\partial x_{i}^{\sigma}}\frac{\partial x_{i}^{\sigma}}{\partial x_{j}^{\nu}}\frac{\partial x_{j}^{\nu}}{\partial x_{j}^{\prime\lambda}}. \)

The **tangent frame bundle** (AKA frame bundle), denoted \({FM}\), is the smooth frame bundle of \({TM}\), i.e. \({(FM,M^{n},GL(n,\mathbb{R}))}\), where the fixed bases in each trivializing neighborhood are again obtained from the atlas on \({M}\), giving the same transition functions as in the tangent bundle. The bases in \({\pi^{-1}(x)}\) are thus defined by

\(\displaystyle e_{p\mu}=f_{i}(p)^{\lambda}{}_{\mu}\frac{\partial}{\partial x_{i}^{\lambda}}. \)

A section of the frame bundle is a frame on \({M}\), and a global section is a global frame, so that \({M}\) is parallelizable iff \({FM}\) is trivial. The right action of a matrix \({g^{\mu}{}_{\lambda}\in GL(n,\mathbb{R})}\) operates on bases as row vectors, and an automorphism of \({FM}\) along with a redefinition of fixed bases to preserve identity sections generates changes of frame in each trivializing neighborhood that preserve the transition functions.

Δ The tangent frame bundle is also denoted \({F(M)}\), but rarely \({F(TM)}\), which is what would be consistent with general frame bundle notation. |

The tangent frame bundle is special in that we can relate its tangent vectors to the elements of the bundle as bases. Specifically, we define the **solder form** (AKA soldering form, tautological 1-form, fundamental 1-form), as a \({\mathbb{R}^{n}}\)-valued 1-form \({\vec{\theta}_{P}}\) on \({P=FM^{n}}\) which at a point \({p=e_{p}}\) projects its argument \({v\in T_{p}FM}\) down to \({M}\) and then takes the resulting vector’s components in the basis \({e_{p}}\), i.e.

\(\displaystyle \vec{\theta}_{P}(v)\equiv\mathrm{d}\pi(v)_{p}^{\mu}. \)

The projection makes the solder form horizontal, and it is also not hard to show it is equivariant, since both actions essentially effect a change of basis:

\(\displaystyle g^{*}\vec{\theta}_{P}(v)=\check{g}^{-1}\vec{\theta}_{P}(v). \)

The pullback by the identity section

\(\displaystyle \begin{aligned}\vec{\theta}_{i} & \equiv\sigma_{i}^{*}\vec{\theta}_{P}\end{aligned} \)

simply returns the components of the argument in the local basis, and thus is identical to the dual frame \({\vec{\beta}}\) viewed as a frame-dependent \({\mathbb{R}^{n}}\)-valued 1-form as we saw previously. Thus recalling the section on horizontal equivariant forms, the values of \({\vec{\theta}_{P}}\) in the fiber over \({x}\) correspond to a single point in the associated bundle \({TM}\), so that the union of the pullbacks \({\vec{\theta}_{i}}\) can be viewed as a single \({TM}\)-valued 1-form on \({M}\)

\(\displaystyle \vec{\theta}\colon TM\rightarrow TM \)

which identifies, or “solders,” the tangent vectors on \({M}\) to elements in the bundle \({TM}\) associated to \({FM}\) (explaining the alternative name “tautological 1-form”).

Δ The \({TM}\)-valued 1-form \({\vec{\theta}}\) is also sometimes called the solder form, and can be generalized to bundles \({E}\) with more general fibers as \({\theta_{E}(v)\colon TM\rightarrow E}\) or \({\theta_{\sigma_{0}}(v)\colon TM\rightarrow V_{\sigma_{0}}E}\), where in the second case \({\sigma_{0}}\) is a distinguished section (e.g. the zero section in a vector bundle). This is called a soldering of \({E}\) to \({M}\); for example a Riemannian metric provides a soldering of the cotangent bundle to \({M}\). In classical dynamics, if \({M}\) is a configuration space then the solder form to the cotangent bundle is called the Liouville 1-form, Poincaré 1-form, canonical 1-form, or symplectic potential. |

Δ The solder form can also be used to identify the tangent space with a subspace of a vector bundle over \({M}\) with higher dimension than \({M}\). |