Fixed points of local actions of nilpotent Lie groups on surfaces

Abstract

Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\phi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $\phi$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem: If the Dold fixed-point index of ${\phi }_t|U$ is nonzero for sufficiently small $t>0$, then $\mathrm{Fix}(G)\cap K\ne \varnothing$.

References in corpus (3)

• Generalized complex geometry and T-duality
• Smooth actions of Lie groups and Lie algebras on manifolds
• Zero sets of Lie algebras of analytic vector fields on real and complex 2-manifolds

Cited by in corpus (2)

• Common zeroes of families of smooth vector fields on surfaces
• Zero sets of Abelian Lie algebras of vector fields