Horizontal differential form on a principal bundle

Let $\pi:P\to M$ be a principal G-bundle. The vertical subbundle is

V:=\ker(d\pi)\subset TP,

as in vertical subbundle.

For $k\ge0$ , a differential $k$ -form $\alpha\in\Omega^k(P)$ is horizontal if, for every $p\in P$ ,

\alpha_p(v_1,\dots,v_k)=0 \quad\text{whenever at least one }v_i\in V_p.

Relation to connections and basic forms

For degree zero the condition is vacuous: every smooth function is horizontal. Horizontality is only a condition relative to the vertical distribution; it does not require choosing a horizontal distribution. However, once a connection is chosen, horizontal forms can be evaluated on horizontal lifts of tangent vectors (compare horizontal lift).

A form on $P$ is basic exactly when it is horizontal and $G$ -invariant (see invariant differential form). Basic forms are precisely pullbacks of forms on $M$ .

Equivalent characterizations

Equivalently, $\alpha$ is horizontal if

\iota_{X^\#}\alpha = 0\quad\text{for every }X\in\mathfrak g,

where $X^\#$ is the fundamental vector field determined by the right principal action.

Examples

Pullbacks from the base are horizontal. If $\beta\in\Omega^k(M)$ , then $\pi^*\beta$ is horizontal (and in fact basic). This is the standard example coming from pullback of differential forms.

Curvature is horizontal; the connection form is not. For a principal connection with connection 1-form $\omega$ , the curvature 2-form $\Omega$ is horizontal (and $\operatorname{Ad}$ -equivariant), so it is tensorial. If the Lie algebra is nonzero, $\omega$ is not horizontal because it reproduces vertical generators: $\omega(X^\#)=X$ (compare reproduction property). See curvature 2-form of a principal connection and connection 1-form.

Solder form on the frame bundle. On the frame bundle of a manifold, the solder form is a canonical horizontal 1-form with values in $\mathbb R^n$ ; it vanishes on vertical vectors because it encodes the projection of tangent vectors to the base.