Non-vanishing complex vector fields and the Euler characteristic

TL;DR

This paper explores the relationship between real and complex vector fields on manifolds, clarifying how the existence of nowhere zero complex vector fields differs from real ones and their implications for the Euler characteristic.

Contribution

It provides a detailed analysis of the conditions under which complex vector fields exist on manifolds, contrasting with real vector fields and their topological restrictions.

Findings

Complex vector fields exist on all manifolds.

Real vector fields are restricted by the Euler characteristic.

The paper clarifies the topological implications of complex vector fields.

Abstract

The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified.