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

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.

v2-changes following the referee's reports. Streamlined Sec. 2 and 3 for better exposition. Corrected the typo in the identification of the kernel of the modified Dirac operator in Sec. 3. Added various remarks throughout the paper for clarity and added more details to the calculations in Sec. 5 on Aloff--Wallach spaces. Updated references and corrected various typos. 31 pages

References in corpus (3)

• Deformation theory of nearly Kähler manifolds
• Minimal Hypersurfaces in nearly ${\mathrm{G}}_2$ Manifolds
• Deformations of asymptotically conical Spin(7)-manifolds