Geometry over R, C, H and O

We develop a unifed theory to study geometry of manifolds with different holonomy groups.

They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special/orientable or not. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively.

For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) A/2-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds.