Induced differential forms on manifolds of functions

Differential forms on the Fréchet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a constant function) and the bar map associating p-forms on F(S,M) to p-forms on M (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians.

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