Second order deformations of associative submanifolds in nearly parallel $G_2$-manifolds

Abstract

Associative submanifolds $A$ in nearly parallel $G_2$-manifolds $Y$ are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over $Y$ has the holonomy group contained in $\mathrm{Spin}(7)$ and the Riemannian cone over $A$ is a Cayley submanifold. Infinitesimal deformations of associative submanifolds were considered by the author. This paper is a continuation of the work. We give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order explicitly. As an application, we show that the infinitesimal deformations of a homogeneous associative submanifold in the 7-sphere given by Lotay, which he called $A_3$, are unobstructed to second order.

28 pages, v2: minor corrections, published version

References in corpus (1)

• Deformation theory of nearly Kähler manifolds

Cited by in corpus (1)

• Special Lagrangians in nearly Kähler ${\mathbb{CP}}^3$