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Deformation theory of nearly $\mathrm{G}_2$ manifolds

Shubham Dwivedi, Ragini Singhal

Abstract

We study the deformation theory of nearly $\mathrm{G}_2$ manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $\mathrm{G}_2$ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $\mathrm{G}_2$ structure on the Aloff--Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly $\mathrm{G}_2$ manifolds.

Deformation theory of nearly $\mathrm{G}_2$ manifolds

Abstract

We study the deformation theory of nearly manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly structure on the Aloff--Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly manifolds.

Paper Structure

This paper contains 11 sections, 22 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $\mathrm{G}_2$ geometry
  3. First order differential operators
  4. Identities for $2$-forms and $3$-forms
  5. Hodge theory of nearly $\mathrm{G}_2$ manifolds
  6. Dirac operators on nearly $\mathrm{G}_2$ manifolds
  7. Harmonic $2$-forms and $3$-forms on nearly $\mathrm{G}_2$ manifolds
  8. Deformations of nearly $\mathrm{G}_2$ structures
  9. Infinitesimal deformations
  10. Second-order deformations
  11. Deformations on the Aloff-Wallach space

Key Result

Proposition 2.4

Suppose $\varphi$ be a $\mathrm{G}_2$ structure on $M$ with $\psi=*\varphi$. Let $\xi$ be a $3$-form which has sufficiently small pointwise norm with respect to $g_{\varphi}$ so that $\varphi+\xi$ is still a positive $3$-form and $\eta$ be a $4$-form with small enough pointwise norm so that $\psi+\e

Theorems & Definitions (56)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Remark 2.7
  • proof
  • Corollary 2.8
  • ...and 46 more