# Curvature of principal bundle

Let $$(P,M,G)$$ be a principal bundle with connection 1-form $$\omega$$. In all books I have seen so far, the curvature is defined by $$\begin{equation} F:=D_{\omega}\omega \in \Omega({P,\mathfrak{g}}) \end{equation}$$ with $$\begin{equation} (D_{\omega}\omega)_p(X_1,...,X_k)=d \omega_p(X_1^H,...,X_k^H) \end{equation}$$ where $$X_i^H$$ means the projection on the horizontal part. However wikipedia (https://en.wikipedia.org/wiki/Gauge_theory_(mathematics)) states, that the curvature can be also see as $$\begin{equation} F(X_1,X_2)=[X_1^\#,X_2^\#]-[X_1,X_2]^\# \in \Omega(M,ad(P)) \end{equation}$$ where # denotes the horizontal lift of the vector fields. What is the exact connection of the two definitions. How can I derive them from each other. I would also be very grateful if someone could provide a reference where the definition wikipedia uses is discussed.

• Could you give a specific reference for your first version? May 13, 2021 at 6:47
• E.g. Mathematical gauge theory by Hamilton May 13, 2021 at 11:25

I'll use $$\Omega \in \Omega^2(P,\mathfrak{g})$$ to denote the curvature tensor of $$\omega$$. One way of identifying these two expressions is through Cartan's structure equation $$\Omega = d\omega + \frac{1}{2}\omega \wedge \omega.$$ A reference is Kobayashi-Nomitsu's book, Chapter II.5; here we use the convention $$d \omega(X,Y) = \frac{1}{2}(X \cdot \omega(Y) - Y \cdot \omega(X) - \omega([X,Y]))$$ where $$X$$ and $$Y$$ are local sections of $$TP$$.
Since $$\Omega$$ is horizontal and $$\omega$$ sends horizontal sections to $$0$$, we have $$\Omega(X,Y) = \Omega(X^H,Y^H) = -\frac{1}{2}\omega([X^H,Y^H]).$$ As $$[X^H,Y^H]^H = [X,Y]^H$$ and $$\omega$$ sends any local section to its vertical part, we have $$\omega([X^H,Y^H]) = [X^H,Y^H] - [X^H,Y^H]^H = [X^H,Y^H] - [X,Y]^H.$$ Finally, you can compare $$\Omega(X,Y)$$ with $$F(X,Y)$$ in your question through the canonical isomorphism from the subspace of $$\Omega^2(P,\mathfrak{g})$$ consisting of $$G$$-equivariant horizontal 2-forms to $$\Omega^2(M,ad(P))$$.
• But $D_\omega\omega$ of the OP seems to be not a 2-form? As written, it has values on $k$-tuples of vector fields for all $k$ May 13, 2021 at 6:46
• @მამუკაჯიბლაძე It is a 2 form. I just wrote down the general expression for $D_{\omega} \omega$ if $\omega$ is a $k$-form May 13, 2021 at 11:26
• @HYL Thanks. That's what I was looking for. I am just a bit confused. The expression for $\Omega$ then needs a $-1/2$ factor which is missing in wikipedia. May 13, 2021 at 11:34
• There are two conventions defining $d \omega$ for a $k$-form $\omega$, according to whether there is a factor $\frac{1}{k+1}$ or not. In Kobayashi-Nomitsu's book, the factor $\frac{1}{k+1}$ is in the definition, this is why we have $\frac{1}{2}$ in the structure equation.
• Thanks. Then there is maybe a mistake in the wikipedia article, because they also use the $1/2$ in the structure equation. May 13, 2021 at 13:17