Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds

Morris Hirsch

doi:10.3390/axioms4030313

Abstract

I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.

Highlights

Classical results of Poincaré [1] (1885), Hopf [2] (1925) and Lefschetz [3] (1937) yield the archetypal fixed point theorem for Lie group actions: Theorem 1
The earliest papers I have found on fixed points for actions of other non-discrete Lie group are those of P
The index i(Ψ, U ) of Ψ in U is defined as the fixed point index of Ψt |U : U → M for sufficiently small t > 0, as defined by Dold [8]

Summary

Introduction

Classical results of Poincaré [1] (1885), Hopf [2] (1925) and Lefschetz [3] (1937) yield the archetypal fixed point theorem for Lie group actions: Theorem 1. Every flow on a compact manifold of non-zero Euler characteristic has a fixed point. The earliest papers I have found on fixed points for actions of other non-discrete Lie group are those of P. If H is a solvable, irreducible affine algebraic group over an algebraically closed field K, every algebraic action of H on a complete algebraic variety over K has a fixed point. Over the field of complex numbers, completeness is equivalent to compactness in the classical topology, and complete nonsingular varieties are compact Kähler manifolds. Sommese [7] extended Borel’s theorem to solvable holomorphic actions on compact. In contrast to the results below, these have no explicit restrictions on dimensions or Euler characteristics

Actions and Local Actions

Fixed Points of Local Actions on Surfaces

Indices of Vector Fields