Grassmann Manifold G(2,8) and Complex Structure on S⁶

Original · EN

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 orthogonal complex structures on R⁸. In this way, we show that G(2,8) and CP³ 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 orthogonal complex structure on the sphere S⁶.