  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 or not. Specialty is an orientation with
respect to the corresponding normed algebra A. 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.
  We also discuss geometric dualities from this viewpoint: Fourier
transformations on A-geometry for flat tori and a conjectural SYZ mirror
transformation from (special) A-geometry to (special) A/2-Lagrangian geometry
on mirror special A-manifolds.
