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

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 $\mathbb M_1(c_1) \times \mathbb M_1(c_1)$andtotallyumbilicalhypersurfacesof$\mathbb M_1(c_1) \times \mathbb M_2(c_2)$($c_1 \neq c_2$) having a local structure product, are of constant mean curvature.