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

TL;DR

This paper characterizes hypersurfaces in four-dimensional product spaces using two distinct Spin$^c$ structures with parallel spinors, linking spinor fields to isometric immersions and properties like constant mean curvature.

Contribution

It introduces a novel approach using two different Spin$^c$ structures to characterize hypersurfaces in product spaces, providing new insights into their geometric properties.

Findings

Characterization of hypersurfaces via parallel spinors

Identification of conditions for constant mean curvature

Application to totally umbilical hypersurfaces

Abstract

The Riemannian product $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$, where $\mathbb M_i(c_i)$ denotes the $2$-dimensional space form of constant sectional curvature $c_i \in \mathbb R$, 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 $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$. As an application, we prove that totally umbilical hypersurfaces of $\mathbb M_1(c_1) \times \mathbb M_1(c_1)$ and totally umbilical hypersurfaces of $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ ($c_1 \neq c_2$) having a local structure product, are of constant mean curvature.