Characterization of hypersurfaces in four-dimensional product spaces via two different Spinc 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 $${\mathrm{Spin}^{\mathrm{c}}}$$ structures carrying each a parallel spinor. restriction these spinor fields to 3-dimensional hypersurface M characterizes isometric immersion into . As an application, we prove that totally umbilical hypersurfaces {\mathbb{M}}_1(c_1)$$ and ( $$c_1 \ne c_2$$ ) having local structure are mean curvature.