Invariant currents by a proper action
Let \(G\) be a connected Lie group acting on a differentiable vector bundle \(E\rightarrow V\), such that the induced action on \(V\) is proper. The author obtains a characterization of the invariant distribution sections of \(E\) by the action of \(G\). The topological vector space \((C^\infty_c(E))_G'\) of the \(G\)-invariant linear functionals on the space \(C_c^\infty (E)\) of \(C^\infty\)-sections with compact support, equipped with the induced weak topology (resp. the strong topology), is isomorphic to the weak (resp. strong) dual of the space \({\overline C}_G^\infty(E)\) of the \(G\)-invariant sections \(\sigma\) with compact quotient supp\((\sigma /G)\) equipped with a suitable topology. If the orbit space is compact, this topology coincides with the usual \(C^\infty\)-topology. If \(G\) is compact, this topology coincides with the Schwartz topology.
differentiable actions on vector bundles
scientific article; zbMATH DE number 1295392