The Energy-Momentum tensor on $Spin^c$ manifolds

TL;DR

This paper investigates the properties of the Energy-Momentum tensor on $Spin^c$ manifolds, deriving formulas for hypersurfaces, metric variations, and characterizing special spinors related to the tensor.

Contribution

It introduces a spinorial Gauss formula, derives a variation formula for the Dirac operator, and characterizes $Spin^c$ Killing spinors via parallel spinors.

Findings

Energy-Momentum tensor appears as second fundamental form

Derived a variation formula for the Dirac operator

Characterized special spinors as restrictions of parallel spinors

Abstract

On $Spin^c$ manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a $Spin^c$ manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized $Spin^c$ Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.