What does covariantly constant mean in the context of $G$-equivariant fiber bundle diffeomorphism? Suppose $M$ be a base (smooth) manifold and $P$ be a Principal $G$-bundle (with $\rho$ as a right $G$-action on it.).
Let $\varphi : P \to P$ be a fiber bundle diffeomorphism which is $G$-equivariant, i.e., $\varphi \circ \pi = \pi \circ \varphi$ and $\rho(g) \circ \varphi = \varphi \circ \rho(g)$ for any $g \in G$.
For given connection $A \in \Omega^1(P;{\frak g})$, what does it mean that $\varphi$ is covariantly constant with respect to the connection $A$? I found this in Wikipedia and I've stuck at here for a long time.
I think it means that $\ker A \subseteq \ker {\rm d} \tilde{\varphi}$ where $\tilde{\phi}:P \to G$ is defined by $\varphi(\rho(g)(p)) = \rho(\tilde{\varphi}(p))(p)$ with property $$ g^{-1}\tilde\varphi(p)g = \tilde\varphi(\rho(g)(p)),$$
but there is no references to support this idea. It seems correct if $G$ is Abelian.
 A: Usually, "being covariantly constant" is used for sections $s$ in vector bundles $E$ which carry a covariant derivative $\nabla$, and then in the obvious way $\nabla s=0$.
The way the Wiki refers to gauge transformations is, I think, a bit different from yours. Denote the following groups:
$$\begin{aligned}
G_1(P)&=\{\varphi:P\to P\,|\,\varphi~G\text{-eqivariant and fiber-preserving bundle isomorphism}\,\},\\
G_2(P)&=\{f:P\to M\,|\,f(p.g)=g^{-1}f(p)g~\text{for all}~p\in P,~g\in G\,\},\\
H(E)&=\{\psi:E\to E\,|\,\psi~ \text{fiber-preserving vector bundle isomorphism}\,\},
\end{aligned}$$
where $E:=P\times_{(G,\Theta)}W$ is the vector bundle associated to $P$ via some representation $\Theta$ of $G$ on some space $W$. Then there exist an isomorphism $G_1\to G_2$ which is basically your map $\varphi\mapsto\tilde{\varphi}$, causing the elements of both groups to be called gauge transformations. Moreover, there exists a homomorphism $G_1\to H$ (which is not surjective in general), and the automorphisms of $E$ in the range of this homomorphism are called gauge transformations as well. As automorphisms of $E$ they are regardable as a section in $\text{End}(E)$. A connection form $\mathcal{A}$ on $P$ induces a covariant derivative $\nabla$ on $\text{End}(E)$ and a covariantly constant gauge transformation is now a gauge transformation such that its $\Gamma\big(\text{End}(E)\big)$-version's covariant derivative vanishes.
I think one can pull this notion back to the principal bundle side and say that $\varphi\in G_1$ is covariantly constant if $\varphi^*\mathcal{A}=\mathcal{A}$, and this identity implies that the related gauge transformation of an associated bundle is covariantly constant. Without thinking about it too long, I think the condition $\varphi^*\mathcal{A}=\mathcal{A}$ is equivalent to your criterion, that is, a gauge transformation $\varphi\in G_1$ preserves the connection form if its $G_2$-version $\tilde{\varphi}$ is constant in the direction of a horizontal tangent vector of $P$.
Maybe this notion of constant on $P$ is a good one. However, I have just seen it before in the vector bundle context, and since physicists barely think about the principal bundle perspective on gauge theory, I guess this is also what the Wiki refers to.
