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].

In this paper we study isometry-invariant Finsler metrics on inner product spaces over $ℝ$ or $ℂ$, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most general concept of the Finsler metric is considered without any additional assumptions that are usually built into its definition. However, we present refined versions of the described results for more specific...