On p-tuples of the Grassmann manifolds

On p-tuples of the Grassmann manifolds

We provide a matrix invarariant for isometry classes of p-tuples of points in the Grassmann manifold Gn K (K = R or C). This invariant fully caracterizes the p-tuple. We use it to determine the regular p-tuples of G2 R , G3 R and G2 C . 1 Introduction and notation A triangle (triple of points) of the Euclidean space is fully de…ned, up to isometry, by three numbers, namely its side lengths. Of ...

Characteristic Classes on Grassmann Manifolds

In this paper, we use characteristic classes of the canonical vector bundles and the Poincaré dualality to study the structure of the real homology and cohomology groups of oriented Grassmann manifold G(k, n). Show that for k = 2 or n ≤ 8, the cohomology groups H∗(G(k, n),R) are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poinc...

Convexity on Affine Grassmann Manifolds

Since the parametrizing space for k-flats in R is the “affine Grassmannian”, G ′ k,d, whose points represent k-flats and whose topology is inherited from that of R in the natural way (a neighborhood of the k-flat spanned by points x0, . . . , xk in general position consisting of all k-flats spanned by points y0, . . . , yk with yi in a neighborhood of xi for each i), what we are asking is wheth...

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 △ 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 comple...

Space Forms of Grassmann Manifolds