Harmonic basis vector fields on surfaces

Abstract

In this paper, we study local parameterizations of surfaces in the Euclidean space $\mathbb{R}^3$ whose coordinate vector fields are harmonic sections with respect to the induced Riemannian metric. After introducing the notion of harmonic basis vector fields on a surface, we derive necessary and sufficient conditions under which the coordinate vector fields $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial v}$ are harmonic. We then focus on two important classes of surfaces: surfaces of revolution and translation surfaces. For these families, we obtain a complete classification of all parameterizations admitting harmonic basis vector fields.