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arXiv · 1610.07864

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

Kotaro Kawai
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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 ${\rm Spin(7)}$ 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.

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Kotaro Kawai. 2016-10-25. Second order deformations of associative submanifolds in nearly parallel $G_2$-manifolds. https://doi.org/10.1093/qmath%2Fhax038

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