Quadratic closed G2-structures
A closed G2-structure is a certain type of geometric structure on a 7-manifold M, given by a 'non-degenerate' closed 3-form. The local geometry of closed G2-structures is non-trivial, in contrast to the perhaps more familiar case of symplectic structures (where we instead have a non-degenerate closed 2-form). In particular, any closed G2-structure automatically induces a Riemannian metric on M. I will talk about a special class of closed G2-structures, those satisfying a further 'quadratic' condition. This is a second order PDE system first written down by Bryant that can be interpreted as a condition on the Ricci curvature of the induced metric. I will focus mainly on the case where the G2-structure is 'extremally Ricci-pinched', giving new examples and describing an unexpected relationship with maximal submanifolds in a certain negatively curved pseudo-Riemannian symmetric space.