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Riemannian Metrics and Harmonic Sections of Spinor Bundles

Simone Farinelli

Abstract

We study the clustering of the lowest non negative eigenvalue of the Dirac operator on a general Dirac bundle when the metric structure is varied. In the classical case we show that any closed spin manifold of dimension greater than or equal to four has a Riemannian metric admitting non trivial harmonic spinors.

Riemannian Metrics and Harmonic Sections of Spinor Bundles

Abstract

We study the clustering of the lowest non negative eigenvalue of the Dirac operator on a general Dirac bundle when the metric structure is varied. In the classical case we show that any closed spin manifold of dimension greater than or equal to four has a Riemannian metric admitting non trivial harmonic spinors.

Paper Structure

This paper contains 13 sections, 20 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dirac bundles
  4. Spectral Resolutions
  5. Spectral Estimates
  6. Warped products and Variations of metrics
  7. Warped Products
  8. Continuity of the Eigenvalues
  9. Differentiability of the Eigenvalues and First Variation
  10. Berger Metrics
  11. Proof of the Main Theorem
  12. Conclusion
  13. Some Results from Functional Analysis

Key Result

Theorem 1.1

Let $(M,P)$ be a closed connected spin manifold of dimension $m\ge4$. Then, there exists a Riemannian metric $g$ on $(M,P)$ admitting non-trivial harmonic spinors.

Theorems & Definitions (34)

  • Conjecture
  • Theorem 1.1
  • Definition 1
  • Proposition 2.1: see e.g. Ba96
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4: Spectral Upper Bounds
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 24 more