2 A ug 2 00 6 Grassmann Manifold G ( 2 , 8 ) and Complex Structure on S 6

In this paper, we use Clifford algebra and the spinor calculus to study the complex structures on Euclidean space R8 and the spheres S4, S6. By the spin representation of G(2, 8) ⊂ Spin(8) we show that the Grassmann manifold G(2, 8) can be looked as the set of complex structures on R8. In this way, we show that G(2, 8) and CP 3 can be looked as twistor spaces of S6 and S4 respectively. Then we show that there is no almost complex structure on sphere S4 and there is no complex structure on the sphere S6.