Dimension des orbites d'une action de \({\mathbb{R}}^ p\) sur une variété compacte. (The dimension of orbits of \({\mathbb{R}}^ p\)-actions on compact manifolds)
Let M be a compact, connected m-manifold without boundary. The rank of M is the largest number of commuting, everywhere linearly independent vector fields on M and is denoted rk(M). If \(\phi\) : \(R^ p\times M\to M\) is a smooth action, the authors define m(\(\phi)\) to be the minimum orbit dimension and relate this number to the rank rk(M). Precisely, they prove the following: Theorem. Let \(M\neq T^ m\) be a compact, connected m-manifold, \(\partial M=\emptyset\), and let \(rk(M)=k\). Then every \(C^{\infty}\) action \(R^ p\times M\to M\) has some orbit of dimension strictly less than \((m+k)/2.\) In particular, \(m(\phi)<(m+k)/2.\)
commuting, everywhere linearly independent vector fields
scientific article; zbMATH DE number 4098321