In this article, we describe various geometries on Riemannian manifolds via a unified approach using normed division algebras and vector cross products. They include K¨ahler geometry, Calabi-Yau geometry, hyperk¨ahler geometry, G2-geometry and geometry of Riemannian symmetric spaces.

Suppose A and B are normed division algebras, i.e. R,C,H or O, we introduce and study Grassmannians of linear subspaces in (A ⊗ B)n which are complex/Lagrangian/maximal isotropic with respect to natural two tensors on (A ⊗ B)n. We show that every irreducible compact symmetric space must be one of these Grassmannian spaces, possibly up to a finite cover. This gives a simple and uniform description of all compact symmetric spaces. This generalizes the Tits magic square description for simple Lie algebras to compact symmetric spaces.

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For two nearby disjoint coassociative submanifolds C and C′ in a G2-manifold, we construct thin instantons with boundaries lying on C and C′ from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.