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

TL;DR

This paper extends bounds for the eigenvalues of the Spin$^c$ Dirac operator on hypersurfaces, explores the existence of Killing spinors on certain manifolds, and characterizes conditions for immersions into complex space forms.

Contribution

It provides new eigenvalue bounds for the Spin$^c$ Dirac operator and establishes a link between Killing spinors and isometric immersions into complex space forms.

Findings

Eigenvalue bounds for the Spin$^c$ Dirac operator on hypersurfaces

Non-existence of totally umbilic surfaces in certain manifolds

Conditions for immersing 3D Sasaki manifolds into complex space forms

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 $\mathbb E^*(\kappa, \tau)$ with 4-dimensional isometry group and isometric immersions of $\mathbb E^*(\kappa, \tau)$ into the complex space form $\mathbb M^4(c)$ of constant holomorphic sectional curvature $4c$, for some $c\in \mathbb R^*$. As applications, we show the non-existence of totally umbilic surfaces in $\mathbb E^*(\kappa, \tau)$ and we give necessary and sufficient geometric conditions to immerse a 3-dimensional Sasaki manifold into $\mathbb M^4(c)$.