Kotaro Kawai

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 introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the phi-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the phi-volume to obtain the stability result in some n-Einstein Sasakian manifolds. It also implies the convexity of the phi-volume functional on the space of affine Legendrian submanifolds. Next, we introduce the notion of special affine Legendrian submanifolds in Sasaki Einstein manifolds as a generalization of that of special Legendrian submanifolds. Then we show that the moduli space of compact connected special affine Legendrian submanifolds is a smooth Frechet manifold. (C) 2016 Elsevier B.V. All rights reserved.

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.

The squashed 7-sphere S-7 is a 7-sphere with an Einstein metric given by the canonical variation and its cone R-8 - {0} has full holonomy Spin(7). There is a canonical calibrating 4-form Phi on R-8 - {0}. A minimal 3-submanifold in S-7 is called associative if its cone is calibrated by Phi. In this paper, we classify two types of fundamental associative submanifolds in the squashed S-7. One is obtained by the intersection with a 4-plane and the other is homogeneous. Then we study their infinitesimal associative deformations and explicitly show that all of them are integrable.

A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone<br /> $C(Y) = \mathbb{R}_{&gt;0} \times Y$ has the holonomy group contained in ${\rm<br /> Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor.<br /> We have a special class of calibrated submanifolds called Cayley submanifolds<br /> in $C(Y)$. An associative submanifold in $Y$ is a minimal 3-submanifold whose<br /> cone is Cayley. We study its deformations, namely, Cayley cone deformations,<br /> explicitly when it is homogeneous in the 7-sphere $S^{7}$.

Coassociative submanifolds are 4-dimensonal calibrated submanifolds in<br /> $G_{2}$-manifolds. In this paper, we construct explicit examples of<br /> coassociative submanifolds in $\Lambda^{2}_{-} S^{4}$, which is the complete<br /> $G_{2}$-manifold constructed by Bryant and Salamon. Classifying the Lie groups<br /> which have 3- or 4-dimensional orbits, we show that the only homogeneous<br /> coassociative submanifold is the zero section of $\Lambda^{2}_{-} S^{4}$ up to<br /> the automorphisms and construct many cohomogeneity one examples explicitly. In<br /> particular, we obtain examples of non-compact coassociative submanifolds with<br /> conical singularities and their desingularizations.