Curvature of principal bundle

Question

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.

Answer 1

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 $\Omega^2(M,ad(P))$.