The Spinc Dirac operator on hypersurfaces and applications

We extend to the eigenvalues of the hypersurface Spinc Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Furthermore, we prove a correspondence between the existence of a Spinc Killing spinor on homogeneous 3-dimensional manifolds E⁎(κ,τ)E⁎(κ,τ) with 4-dimensional isometry group and isometric immersions of E⁎(κ,τ)E⁎(κ,τ) into the complex space form M4(c)M4(c) of constant holomorphic sectional curvature 4c  , for some c∈R⁎c∈R⁎. As applications, we show the non-existence of totally umbilic surfaces in E⁎(κ,τ)E⁎(κ,τ) and we give necessary and sufficient geometric conditions to immerse a 3-dimensional Sasaki manifold into M4(c)M4(c).