Critical points of invariant functions on closed orientable surfaces

Abstract

The relationship between critical points of equivariant functions and topological invariants of an equivariant action on closed manifold is an interesting problem. In this paper, we study this relationship for orientation-preserving actions of finite groups $$G$$ on a closed orientable surfaces. We give an elementary, but detailed, description of the behaviour of the gradient field of an equivariant $$C^1$$ -function, we present an elementary, differential, proof of the Riemann–Hurwitz formula and we construct invariant $$C^1$$ -functions with the minimal number of critical orbits. These lead us to show that, with a few exceptions, the equivariant Lusternik–Schnirelmann category of a closed orientable topological surface equals the number of singular orbits of the action.