TITLE: Exterior Derivative of One-Form vs Torsion of Connection
QUESTION [8 upvotes]: Let $\omega$ be a $1$-form. Then $d\omega$ may be defined by the formula
$$
d\omega(X,Y) = \frac{\partial}{\partial X}\iota_Y\omega - \frac{\partial}{\partial Y}\iota_X\omega-\omega([X,Y])
$$
where $X,Y$ are vector fields. This formula bears a resemblance to the formula for the torsion of a connection $\nabla$:
$$
\nabla_XY - \nabla_YX - [X,Y]
$$
Is there a geometric explanation for this resemblance?

REPLY [2 votes]: Note a very deep result, but I found a common pattern and that the third term cancels some terms.
You gave two expressions and I found a third on Wikipedia:
$$\begin{align}
d\omega(X,Y) &= \partial_X \iota_Y \omega - \partial_Y \iota_X \omega - \omega([X,Y]) 
\\
T(X,Y) &= \nabla_X Y - \nabla_Y X - [X,Y]
\\
R(X,Y) &= \nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]}
\end{align}$$
They all have a common form,
$$
\Omega(X,Y) = D(X) A(Y) - D(Y) A(X) - A([X,Y]),
$$
where $D(X)=\partial_X+\Gamma_X$ for some automorphism $\Gamma_X$ (possibly vanishing). Below, $D_j=\partial_j+\Gamma_j$ will be used for $D(\partial_j).$ Also, $A_j=A(\partial_j).$
For the three cases we have
$$\begin{align}
d\omega(X,Y) &: D(X)=\partial_X,\ A(X)=\iota_X\omega = \omega(X)
\\
T(X,Y) &: D(X)=\nabla_X,\ A(X)=X
\\
R(X,Y) &: D(X)=\nabla_X,\ A(X)=\nabla_X
\end{align}$$
With $X=X^j\partial_j,\ Y=Y^k\partial_k$ we have
$$\begin{align}
D(X) A(Y)
&= D(X^j\partial_j) A(Y^k\partial_k)
= X^j D(\partial_j) (Y^k A(\partial_k))
\\
&= X^j (\partial_j+\Gamma_j) (Y^k A_k) \\
&= X^j ((\partial_j Y^k) A_k + Y^k (\partial_j A_k) + Y^k \Gamma_j A_k) \\
&= X^j (\partial_j Y^k) A_k + X^j Y^k (D_j A_k)
\\
D(Y) A(X) 
&= Y^k (\partial_k X^j) A_j + X^j Y^k (D_k A_j)
\\
A([X,Y]) &= A(X^j(\partial_j Y^k)\partial_k - Y^k(\partial_k X^j)\partial_j)
\\
&= X^j(\partial_j Y^k) A_k - Y^k (\partial_k X^j) A_j
\end{align}$$
and when we then simplify $\Omega(X,Y)$ we see that $A([X,Y])$ cancels two terms from $D(X) A(Y) - D(Y) A(X)$:
$$
D(X) A(Y) - D(Y) A(X) - A([X,Y])
\\
= \left( X^j (\partial_j Y^k) A_k + X^j Y^k (D_j A_k) \right)
- \left( Y^k (\partial_k X^j) A_j + X^j Y^k (D_k A_j) \right)
- \left( X^j(\partial_j Y^k) A_k + Y^k (\partial_k X^j) A_j \right)
\\= X^j Y^k (D_j A_k - D_k A_j)
$$

EDIT 15 July 2021
A deeper explanation is that all three of these are outer derivatives.
We are used to handle scalar-valued forms, but what if the form is vector-valued? Then we use the exterior covariant derivative, which I assume (without thorough investigation) can be written
$$
D\omega(X,Y) = \nabla_X\omega(Y) - \nabla_Y\omega(X) - \omega([X,Y]).
$$
Taking $\omega(X)=X$ makes $D\omega$ be the torsion, and taking $\omega_Z(X)=\nabla_X Z$ makes $D\omega$ be the curvature.