Elliptic spaces in Grassmann manifolds

Space Forms of Grassmann Manifolds

Transformations of Grassmann spaces

This is a current version of a part of the book “Transformations of Grassmann spaces”. We study transformations of Grassmann spaces preserving certain geometrical constructions (related with buildings). The next part will be devoted to Grassmann spaces associated with polar spaces.

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...

Cobordism independence of Grassmann manifolds

This paper is a continuation of the ongoing study of cobordism of Grassmann manifolds. Let F denote one of the division rings R of reals, C of complex numbers, or H of quaternions. Let t = dimRF . Then the Grassmannian manifold Gk(F) is defined to be the set of all k-dimensional (left) subspaces of Fn+k. Gk(F) is a closed manifold of real dimension nkt. Using the orthogonal complement of a subs...

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...