The Geometry and Topology on Grassmann Manifolds

This paper shows that the Grassmann Manifolds $G_{\bf F}(n,N)$ can all be imbedded in an Euclidean space $M_{\bf F}(N)$ naturally and the imbedding can be realized by the eigenfunctions of Laplacian $\triangle$ on $G_{\bf F}(n,N)$. They are all minimal submanifolds in some spheres of $M_{\bf F}(N)$ 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.Comment: 15 page

A Note on Characteristic Classes

This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.Comment: 10 page

A Note on the Degenerate Morse Inequalities

In this paper we give an analytic proof of the degenerate Morse inequalities in the spirit of E. Witten. The max-min methods are used to estimate the number of ‘small’ eigenvalues of Witten’s deformed Laplacian.</p

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

In this paper, we use the Clifford algebra C&#8467;8 to construct fibre bundles ¿1 : G(2; 8) &#8594; S6, ¿'1 : G(2; 7) &#8594; S6 and ¿2 : G(3; 8) &#8594; 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.</p