Holomorphic structures for surfaces in Euclidean n-space

2017/07/23 by Katsuhiro Moriya, Moriya, Katsuhiro
Mathematics · #53C27 #53C43 #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Primary: 53C42 #Secondary: 53A30

paper · pdf · doi:10.48550/arxiv.1707.07246

openalex publication_date 2017/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean three- or four-space. The Weierstrass representation, the spin transform, the Darboux transforms, surfaces of parallel mean curvature vector, families of flat connections associated with a harmonic map from a Riemann surface to a sphere are explained. The degree of the spinor bundle associated with a conformal immersion is calculated. Analogues of a polar surface and a bipolar surface of a minimal immersion into a three-sphere are defined. They are shown to be minimal surfaces in a sphere.

Citations

• Harmonic Mappings of Riemannian Manifolds

Related

• The spinor and Weierstrass representations of surfaces in space
• The Spinor Representation of Surfaces in Space
• Clifford structures, bilegendrian surfaces, and extrinsic curvature
• Spinor Representations of Surfaces in 4-Dimensional Pseudo-Riemannian Manifolds
• Surfaces in ℝ⁷ obtained from harmonic maps in S⁶