0
$\begingroup$

A connection form is a Lie-algebra-valued one form $\omega\in\Omega^1(P,\mathfrak{g})$ on a principal bundle P. I have also seen this written this as $\Omega^1(P)\otimes\mathfrak{g}$, but I do not see the reasoning behind this notation.

$\endgroup$
1
$\begingroup$

If $M$ is a manifold and $V$ a vector space, you can think of a "$V$-valued one-form" on $M$ as element of $T^*M \otimes V$ in the following way. If $(x_1,\dots,x_n)$ are local coordinates on $M$ and $e_1,\dots,e_m$ is a basis of $V$, then the one-form which sends the basis vector $\frac{\partial}{\partial x_i}$ (of $TM$) to the basis vector $e_j$ (of V) can be thought of as the element $dx_i \otimes e_j$.

In other words, if $\omega \in T^*M$ is an ordinary one-form, and $v \in V$, and $X \in TM$ a tangent vector, then $\omega \otimes v$ is the $V$-valued one-form which acts on $X$ via

$$ (\omega \otimes v)(X) = \omega(X) \, v $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.