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Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor

Samuel Lockman

TL;DR

The paper develops a comprehensive framework for classifying and understanding imaginary generalized Killing spinors on $ ext{spin}^c$ manifolds, bridging Riemannian and semi-Riemannian geometries. It constructs parallel spinors on leaves of foliations defined by Dirac currents, obtains local and global splitting results via flows of closed (and conformal) vector fields, and applies a spin^c Gauss formula to relate ambient GK spinor data to leaf geometry. The Riemannian classification separates type I from type II imaginary GK spinors, yielding leafwise parallel spinors and local isometries that generalize Baum, Grosse–Nakad, and Leistner–Lischewski results to the $ ext{spin}^c$ context, with type II arising from Lorentzian hypersurface reductions. Collectively, these results illuminate the geometric structure enforced by GK spinors and establish a robust link between spinorial equations, Dirac currents, and foliation-based decompositions with potential implications for holonomy and warped-product constructions.

Abstract

We classify Riemannian $\text{spin}^c$ manifolds carrying a type I imaginary generalized Killing spinor, by explicitly constructing a parallel spinor on each leaf of the canonical foliation given by the Dirac current. We also provide a class of Riemannian $\text{spin}^c$ manifolds carrying a type II imaginary generalized Killing spinor, by considering spacelike hypersurfaces of Lorentzian $\text{spin}^c$ manifolds. We carry out much of the work in the setting of semi-Riemannian $\text{spin}^c$-manifolds carrying generalized Killing spinors, allowing us to draw conclusions in this setting as well. In this context, the Dirac current is not always a closed vector field. We circumvent this in even dimensions, by considering a modified Dirac current, which is closed in the cases when the original Dirac current is not. On the path to these results, we also study semi-Riemannian manifolds carrying closed and conformal vector fields.

Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor

TL;DR

The paper develops a comprehensive framework for classifying and understanding imaginary generalized Killing spinors on manifolds, bridging Riemannian and semi-Riemannian geometries. It constructs parallel spinors on leaves of foliations defined by Dirac currents, obtains local and global splitting results via flows of closed (and conformal) vector fields, and applies a spin^c Gauss formula to relate ambient GK spinor data to leaf geometry. The Riemannian classification separates type I from type II imaginary GK spinors, yielding leafwise parallel spinors and local isometries that generalize Baum, Grosse–Nakad, and Leistner–Lischewski results to the context, with type II arising from Lorentzian hypersurface reductions. Collectively, these results illuminate the geometric structure enforced by GK spinors and establish a robust link between spinorial equations, Dirac currents, and foliation-based decompositions with potential implications for holonomy and warped-product constructions.

Abstract

We classify Riemannian manifolds carrying a type I imaginary generalized Killing spinor, by explicitly constructing a parallel spinor on each leaf of the canonical foliation given by the Dirac current. We also provide a class of Riemannian manifolds carrying a type II imaginary generalized Killing spinor, by considering spacelike hypersurfaces of Lorentzian manifolds. We carry out much of the work in the setting of semi-Riemannian -manifolds carrying generalized Killing spinors, allowing us to draw conclusions in this setting as well. In this context, the Dirac current is not always a closed vector field. We circumvent this in even dimensions, by considering a modified Dirac current, which is closed in the cases when the original Dirac current is not. On the path to these results, we also study semi-Riemannian manifolds carrying closed and conformal vector fields.

Paper Structure

This paper contains 11 sections, 18 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Generalized Killing spinors
  4. The geometry of semi-Riemannian manifolds carrying closed and conformal vector fields
  5. Non-local splittings without completeness assumptions
  6. Global splittings under completeness assumptions
  7. Returning to generalized Killing spinors
  8. The semi-Riemannian $\text{spin}^c$ setting
  9. The Riemannian classification
  10. Type I imaginary generalized Killing spinors
  11. Type II imaginary generalized Killing spinors

Key Result

Lemma 3

Let $\varphi \in \Gamma(S)$ be a generalized Killing spinor and suppose that either Then in Case case-odd-1 and Case case-even-i, $V_{\varphi}$ is a closed vector field with where $c = -2$ if $r \mod{4} \in \{0,3\}$ and $c = 2$ otherwise. In Case case-modified-2 and Case case-modified, $\overline{V}_{\varphi}$ is a closed vector field with If $\varphi$ is furthermore a special generalized Killi

Theorems & Definitions (41)

  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Lemma 4: Kühnel & Rademacher, 1994, Kuhnel1994
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 31 more