  In this paper we study the geometry of manifolds with vector cross product
and its complexification. First we develop the theory of instantons and branes
and study their deformations. For example they are (i) holomorphic curves and
Lagrangian submanifolds in symplectic manifolds and (ii) associative
submanifolds and coassociative submanifolds in G_2-manifolds.
  Second we classify Kahler manifolds with the complex analog of vector cross
product, namely they are Calabi-Yau manifolds and hyperkahler manifolds.
Furthermore we study instantons, Neumann branes and Dirichlet branes on these
manifolds. For example they are special Lagrangian submanifolds with phase
angle zero, complex hypersurfaces and special Lagrangian submanifolds with
phase angle pi/2 in Calabi-Yau manifolds.
  Third we prove that, given any Calabi-Yau manifold, its isotropic knot space
admits a natural holomorphic symplectic structure. We also relate the
Calabi-Yau geometry of the manifold to the holomorphic symplectic geometry of
its isotropic knot space.
