The Geometry On Grassmann Manifold G(2,8)

In this paper, we study the geometry on G(2, 8), the one case of Grassmann manifolds. We consider Grassmann manifold G(2,8) as a submanifold of the unit sphere of /\2(Rn) which is the exterior vector space on Rn. Thus we have the induced metric and connection on G(2, n). We use differential geometry and Clifford algebra to construct a map r from G(2,8) to 56, which makes Grassmann manifold G(2,8) a fibre bundle over the unit sphere S6 and the fibre is the complex projective space CP3. And w,e prove that CP3 and S6 are the homologically volume minimizing submanifolds of G(2,8) by calibration, furthermore, they generate the homological group H6(G(2,8)) of G(2,8).Huanghui(Differential geometry) Directed by: Associate Professor Zhou jianwei...