Principal connections and vertical curvature

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as recorded by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), and let $\pi :P\to M$ be a smooth right principal $G$-bundle in the sense of Smooth principal $G$-bundle. The tangent bundle $TP$ has the canonical smooth structure supplied by Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure. Its vertical subbundle is $VP:=\ker (d\pi )\subseteq TP\mathrm{.}$ In each principal chart $U\times G$, this is the tangent bundle in the $G$ factor, so it is a smooth vector subbundle of $TP$ in the sense of Smooth vector bundles, rank, fibres, and trivial bundles.

A principal connection on $P$ is a smooth vector subbundle $H\subseteq TP$ such that $T_{p}P=V_{p}P\oplus H_p$ for every $p\in P$ and $d(R_g{)}_p(H_p)=H_{pg}(p\in P,g\in G),$ where $R_g(p)=pg$ is the principal right action. Thus $H$ is a smoothly varying horizontal complement to the vertical tangent directions, invariant under the structure-group action. This definition is conditional on a specified $H$; it is not an existence theorem for connections on every bundle.

Let $h:TP\to H$ and $v:TP\to VP$ be the projections given by this direct-sum decomposition. For smooth vector fields $X,Y$ on $P$, define the vertical-valued curvature by $K_H(X,Y):=-v([hX,hY]),$ where the vector-field bracket is the commutator The Lie bracket of smooth vector fields. The bracket definition gives the commutator on functions; Smooth vector fields form a Lie algebra under the Lie bracket proves that it is again a smooth vector field. In a local frame adapted to the smooth splitting $TP=VP\oplus H$, the projections $h$ and $v$ are smooth. Therefore $v([hX,hY])$ is a smooth vertical vector field whenever $X,Y$ are smooth. This formula is tensorial in $X,Y$: for a smooth function $f$, $v([fhX,hY])=fv([hX,hY])-hY(f)v(hX)=fv([hX,hY]),$ because $v(hX)=0$, and the same calculation in the second input gives $v([hX,fhY])=fv([hX,hY])$, since $v(hY)=0$. It is alternating because the bracket is alternating. To see the induced bundle map is smooth, choose local smooth frames $E_1,\dots ,E_n$ of $TP$ and $F_1,\dots ,F_r$ of $VP$. Tensoriality gives, for $X={\sum }_i{f}_i{E}_i$ and $Y={\sum }_j{g}_j{E}_j$, $K_H(X,Y)={\sum }_{i,j}{f}_i{g}_j{K}_H(E_i,E_j)\mathrm{.}$ Each $K_H(E_i,E_j)$ is a smooth section of $VP$, so its components in the $F_a$ frame are smooth; the displayed formula therefore has smooth local coefficient functions. Thus the formula defines a smooth alternating bundle map $TP{\times }_{P}TP\to VP$, independent of the local vector-field extensions representing the tangent vectors.

This item uses the vertical-valued convention. It does not identify $VP$ with a Lie-algebra-valued form or assert a Maurer–Cartan formula. Its axiom base contains ${\mathrm{A}\mathrm{C}}_{\omega }$ exactly through the canonical tangent-bundle and vector-field suppliers above; the displayed splitting and curvature formula themselves select no objects from a family.