The Geometry and Topology on Grassmann Manifolds

This paper shows that the Grassmann Manifolds GF(n,N) can all be imbedded in an Euclidean space MF(N) naturally and the imbedding can be realized by the eigenfunctions of Laplacian Δ on GF(n,N). They are all minimal submanifolds in some spheres of MF(N) respectively. Using these imbeddings, we construct some degenerate Morse functions on Grassmann Manifolds, show that the homology of the complex and quaternion Grassmann Manifolds can be computed easily.