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Characterization of hypersurfaces in four-dimensional product spaces via two different Spinc structures
Annals of Global Analysis and Geometry  (IF0.846),  Pub Date : 2021-10-07, DOI: 10.1007/s10455-021-09802-4
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. 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 $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ ($$c_1 \ne c_2$$) having a local structure product are of constant mean curvature.