Totally umbilical hypersurfaces of Spin$^c$ manifolds carrying special spinor fields

TL;DR

This paper proves that under certain conditions, totally umbilical hypersurfaces in Spin$^c$ manifolds with special spinors have constant mean curvature, extending classical results and showing non-existence in certain cases.

Contribution

It extends Kowalski's classical result to Spin$^c$ manifolds with special spinors and demonstrates non-existence of certain hypersurfaces in non-constant curvature Spin manifolds.

Findings

Totally umbilical hypersurfaces with special spinors have constant mean curvature.

No extrinsic hypersurfaces with special spinors exist in certain complete Spin manifolds.

Extension of classical Einstein manifold results to Spin$^c$ manifolds.

Abstract

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin$^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin$^c$ case the result of O. Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to $3$ is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian Spin manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.