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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 ${\Omega }^p\left(M\right)\to {\Omega }^{p-k}\left(ℱ(S,M)\right)$ (hat pairing with a constant function) and the bar map ${\Omega }^p\left(M\right)\to {\Omega }^p\left(ℱ(S,M)\right)$ (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6].

Summary: Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.