Common zeros of inward vector fields on surfaces

TL;DR

This paper extends the concept of the Poincaré-Hopf index to blocks of zeros of inward vector fields on surfaces, providing conditions for the existence of zeros and applications to Lie algebra actions.

Contribution

It introduces a generalized index for zero blocks of inward vector fields on surfaces and establishes zero existence results under analyticity and area-preserving conditions.

Findings

Y has a zero in K if X and Y are analytic

Y has a zero in K if Y is C^2 and area-preserving

Applications to Lie algebra and group actions

Abstract

A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow $\Phi^X$. A compact relatively open set K in the zero set of X is a block. The Poincar\'e-Hopf index is generalized to an index for blocks that may meet the boundary. A block with nonzero index is essential. Let X, Y be inward $C^1$ vector fields on surface M such that $[X,Y]\wedge X=0$ and let K be an essential block of zeros for X. Among the main results are that Y has a zero in K if X and $Y$ are analytic, or Y is $C^2$ and $\Phi^Y$ preserves area. Applications are made to actions of Lie algebras and groups.