The tangent bundle and solder form

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

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

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 \({x}\) correpond 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 \({M}\).

An Illustrated Handbook