On generalized sections of locally free actions of compact Lie groups (Q2764824)

Let \(M\) be an \((m+n)\)-dimensional compact and orientable manifold. Let \(G\) be a connected and compact \(m\)-dimensional Lie group with the Lie algebra \(\mathfrak g\) and \(\Phi:G\times M\to M\) be a locally free action. The orbits of \(\Phi\) are leaves of a foliation denoted by \(\mathcal F\). An \(n\)-dimensional immersed submanifold \(S\) of \(M\) orthogonal to the orbits of \(\Phi\) is called a generalized section of \(\Phi\). Theorem: If \(\Phi\) admits a generalized section then there exists an isomorphism NEWLINE\[NEWLINEH^{\ast}\left(M,\mathbb{R}\right)\cong H^{\ast}\left({\mathfrak g}\right)\otimes H^{\ast}\left(M/{\mathcal F}\right).NEWLINE\]