vertical differential form



In differential geometry

Let π:PX\pi : P \to X be a bundle in the category Diff of smooth manifolds.

The dg-algebra Ω vert (P)\Omega^\bullet_{vert}(P) of vertical differential forms on PP is the quotient of the de Rham complex dg-algebra Ω (P)\Omega^\bullet(P) of all forms on PP, by the dg-ideal of horizontal differential forms, hence of all those forms that vanish when any one vector in their arguments is a vertical vector field in that it is in the kernel of the differential dπ:TPTXd \pi : T P \to T X.

For a trivial bundle P=X×FP = X \times F the underlying complex of Ω vert (P)\Omega^\bullet_{vert}(P) is C (X×F) Γ(T *F)\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F).


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