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

Roger Nakad (1) , Julien Roth (2)

1 NDU - Notre Dame University-Louaize [Lebanon]
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

The Riemannian product M1(c1)×M2(c2), where Mi(ci) denotes the 2-dimensional space form of constant sectional curvature ci ∈ 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 M1(c1) × M2(c2). As an application, we prove that totally umbilical hypersurfaces of M1(c1) × M1(c1) and totally umbilical hypersurfaces of M1(c1) × M2(c2) (c1 = c2) having a local structure product, are of constant mean curvature.

Mots clés: Spin c structures on hypersurfaces; totally umbilcal hypersurfaces; parallel Spin c spinors; generalized Killing Spin c spinors; Kähler manifolds Mathematics subject classifications (2010): 53C27; 53C40; 53C80

Domaines: Géométrie différentielle [math.DG]

Roger Nakad, Julien Roth. Characterization of hypersurfaces in four dimensional product spaces via two different Spin^c structures. Annals of Global Analysis and Geometry, 2021, 61 (1), pp.89-114. ⟨hal-02304263⟩