Rigidity, stability and deformations in nearly parallel G2 geometry

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In our project we study properties of an important class of 7-dimensional Einstein manifolds: nearly parallel G2-manifolds. This geometry has two main elements, the Einstein metric and the class of associative submanifolds. We plan to obtain a better understanding, in particular by the study of deformations, of these elements and of the geometry itself.The goals of our project can be grouped in three parts.In part A we want to prove the stability of the Einstein metric or to find conditions implying this stability in general.In part B we plan to study deformations of the nearly parallel G2 structure. There is one important example where the existence of deformations still could not be ruled out. We plan to answer this question and based on this to show in general the rigidity (i.e. non existence of deformations) of nearly parallel G2 structures.In part C we plan the study of associative sub manifolds. We aim at a classification under several natural geometric conditions. Also we plan to find new example and to study deformations. Here we want to show rigidity (e.g. under curvature conditions), but also geometric properties, e.g. the preservation of the volume under deformations.