河井 公大朗

In this paper we show that a parallel differential form $\Psi$ of even degree<br /> on a Riemannian manifold allows to define a natural differential both on<br /> $\Omega^\ast(M)$ and $\Omega^\ast(M, TM)$, defined via the<br /> Fr\&quot;olicher-Nijenhuis bracket. For instance, on a K\&quot;ahler manifold, these<br /> operators are the complex differential and the Dolbeault differential,<br /> respectively. We investigate this construction when taking the differential<br /> w.r.t. the canonical parallel $4$-form on a $G_2$- and ${\rm<br /> Spin}(7)$-manifold, respectively. We calculate the cohomology groups of<br /> $\Omega^\ast(M)$ and give a partial description of the cohomology of<br /> $\Omega^\ast(M, TM)$.

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

We extend the characterization of the integrability of an almost complex<br /> structure $J$ on differentiable manifolds via the vanishing of the<br /> Fr\&quot;olicher-Nijenhuis bracket $[J, J] ^{FN}$ to an analogous characterization<br /> of torsion-free $G_2$-structures and torsion-free Spin(7)-structures. We also<br /> explain the Fern\&#039;andez-Gray classification of $G_2$-structures and the<br /> Fern\&#039;andez classification of Spin(7)-structures in terms of the<br /> Fr\&quot;olicher-Nijenhuis bracket.