Coframe

It is dual to frame on a manifold.
It is an ordered set of 1-forms ${θ^{1}, \dots, θ^{m}}$ defined on an open set $U$ of an $m$ -dimensional manifold $M$ such that they span the cotangent space $T_{x} M$ for every $x \in U$ .

An ordered set ${θ^{1}, \dots, θ^{m}}$ define a coframe if and only if

θ^{1} \land \dots \land θ^{m} \neq 0

for every $x \in U$ ([Olver_1995]).

A global coframe provides, therefore, an orientation and a trivialization o $T^{*} M \approx M \times R^{m}$ , so it is also called absolute parallelism.

Generation of k-forms

A coframe ${θ^{1}, \dots, θ^{m}}$ does not only generate any 1-form, but also any differential $k$ -form can be written as linear combination of $k$ -fold wedge products:

\sum_{I} h_{I} (x) θ^{i_{1}} \land \dots \land θ^{i_{k}}

See [Olver_1995] page 254.

Duality with frames

A frame on a manifold and a coframe are dual if they are dual basis for every $x \in U$ . Given a coframe, the dual frame will be denoted by

\frac{\partial}{\partial θ^{j}}

Given coordinates $(x_{1} \dots, x_{m})$ on the manifold, we have distinguished local frame and coframe denoted respectively by

{\frac{\partial}{\partial x^{1}}, \dots, \frac{\partial}{\partial x^{m}}}

and

{d x^{1}, \dots, d x^{m}} .

Example of computation in this coframe: Lie derivative of a 1-form in the canonical coframe.

Given a coframe ${θ^{1}, \dots, θ^{m}}$ any $k$ -form $Ω$ can be written as a linear combination

Ω = \sum_{I} h_{I} (x) θ^{i_{1}} \land \dots \land θ^{i_{k}} .

In particular,

d θ^{i} = \sum_{\binom{j, k = 1}{j < k}}^{m} T_{j k}^{i} θ^{j} \land θ^{k}

expressions which are called structure equations of the coframe and $T_{j k}^{i}$ are called structure coefficients. They are also called Maurer-Cartan equations when we are dealing with a left-invariant coframe on a Lie group. See Maurer-Cartan form#Structural equation/MC equation.

These equations are the dual version of

[\frac{\partial}{\partial θ^{j}}, \frac{\partial}{\partial θ^{k}}] = - \sum_{i = 1}^{m} T_{j k}^{i} \frac{\partial}{\partial θ^{i}}

([Olver 1995]). This relation is proved by the infinitesimal Stokes' theorem.

Visualization

(where we are calling $e_{j}$ to $\frac{\partial}{\partial θ^{j}}$ ):
Pasted image 20211226092246.png

Pasted image 20211226092359.png