# Induced differential forms on manifolds of functions

## Cornelia Vizman

Address: West University of Timişoara, Department of Mathematics

E-mail: vizman@math.uvt.ro

Abstract: Differential forms on the Fréchet manifold $\mathcal{F}(S,M)$ 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 $\Omega ^p(M)\rightarrow \Omega ^{p-k}(\mathcal{F}(S,M))$ (hat pairing with a constant function) and the bar map $\Omega ^p(M)\rightarrow \Omega ^p(\mathcal{F}(S,M))$ (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6].

AMSclassification: primary 11K11; secondary 22C22.

Keywords: manifold of functions, fiber integral, diffeomorphism group.