Geometry on Grassmann Manifolds G(2,8) and G(3,8)

In this paper, we use the Clifford algebra Cℓ₈ to construct fibre bundles ¿₁ : G(2; 8) → S⁶, ¿'₁ : G(2; 7) → S⁶ and ¿₂ : G(3; 8) → S⁷, the fibres are CP³, CP² and ASSOC = G₂=SO(4) respectively. We show that G(2; 5), CP³ and S⁶ are the homologically volume minimizing submanifolds of G(2; 8) by calibrations and they generate the homology group H₆(G(2; 8)). The submanifolds S⁷ and ASSOC of G(3; 8) generate H₇(G(3; 8)) and H₈(G(3; 8)) respectively.