The Spin${}^c$ Dirac Operator on Hypersurfaces and Applications

Abstract

We extend to the eigenvalues of the hypersurface Spin${}^c$ Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Futhermore, we prove a correspondence between the existence of a Spin${}^c$ Killing spinor on homogeneous 3-dimensional manifolds ${\mathbb{E}}^{\ast }(\kappa ,\tau )$ with 4-dimensional isometry group and isometric immersions of ${\mathbb{E}}^{\ast }(\kappa ,\tau )$ into the complex space form ${\mathbb{M}}^4(c)$ of constant holomorphic sectional curvature $4c$, for some $c\in {\mathbb{R}}^{\ast }$. As applications, we show the non-existence of totally umbilic surfaces in ${\mathbb{E}}^{\ast }(\kappa ,\tau )$ and we give necessary and sufficient geometric conditions to immerse a 3-dimensional Sasaki manifold into ${\mathbb{M}}^4(c)$.

References in corpus (2)

• On Eta-Einstein Sasakian Geometry
• The Energy-Momentum tensor on $Spi{n}^c$ manifolds

Cited by in corpus (1)

• Complex Generalized Killing Spinors on Riemannian Spin${}^c$ manifolds