Knowlpedia

Horizontal differential form on a principal bundle

A differential form on a principal bundle that vanishes whenever any input vector is vertical

Let π:P→M\pi:P\to M be a . The vertical subbundle is

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

as in .

A differential kk-form α∈Ωk(P)\alpha\in\Omega^k(P) is horizontal if, for every p∈Pp\in P,

αp(v1,…,vk)=0whenever at least one vi∈Vp.\alpha_p(v_1,\dots,v_k)=0 \quad\text{whenever at least one }v_i\in V_p.

Horizontality is only a condition relative to the vertical distribution; it does not require choosing a . However, once a connection is chosen, horizontal forms can be evaluated on horizontal lifts of tangent vectors (compare ).

A form on PP is exactly when it is horizontal and GG-invariant (see ). Basic forms are precisely pullbacks of forms on MM.

Equivalent characterizations

Equivalently, α\alpha is horizontal if

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

where X#X^\# is the determined by the right principal action.

Examples
  1. Pullbacks from the base are horizontal. If β∈Ωk(M)\beta\in\Omega^k(M), then π∗β\pi^*\beta is horizontal (and in fact basic). This is the standard example coming from .
  1. Curvature is horizontal; the connection form is not. For a principal connection with connection 1-form ω\omega, the curvature 2-form Ω\Omega is horizontal (and Ad⁡\operatorname{Ad}-equivariant), so it is tensorial. By contrast, ω\omega is not horizontal because it reproduces vertical generators: ω(X#)=X\omega(X^\#)=X (compare ). See and .
  1. Solder form on the frame bundle. On the of a manifold, the is a canonical horizontal 1-form with values in Rn\mathbb R^n; it vanishes on vertical vectors because it encodes the projection of tangent vectors to the base.