Characterization of hypersurfaces in four dimensional product spaces via two different Spin^c structures

2019/10/01 by Roger Nakad, Julien Roth, Nakad, Roger +1
Mathematics · #53C27 #53C40 #53C80 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1910.01025

openalex publication_date 2019/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Riemannian product \mathbb M₁(c₁) × \mathbb M₂(c₂), where \mathbb M_i(c_i) denotes the 2-dimensional space form of constant sectional curvature c_i ∈ \mathbb R, has two different Spin^c structures carrying each a parallel spinor. The restriction of these two parallel spinor fields to a 3-dimensional hypersurface M characterizes the isometric immersion of M into \mathbb M₁(c₁) × \mathbb M₂(c₂). As an application, we prove that totally umbilical hypersurfaces of \mathbb M₁(c₁) × \mathbb M₁(c₁) and totally umbilical hypersurfaces of \mathbb M₁(c₁) × \mathbb M₂(c₂) (c₁ ≠ c₂) having a local structure product, are of constant mean curvature.

Citations

• Surfaces in S³ and H³ via Spinors
• Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy
• A Bonnet theorem for isometric immersions into products of space forms
• Isometric immersions into products of space forms
• An Alexandrov theorem in Minkowski spacetime

Related

• Totally umbilical hypersurfaces of Spin^c manifolds carrying special\n spinor fields
• Spinorial representation of submanifolds in a product of space forms
• On complete submanifolds with parallel mean curvature in product spaces
• Surfaces with parallel mean curvature in ℂPⁿ×ℝ and ℂHⁿ×ℝ
• Isoparametric hypersurfaces in product spaces