Volume 9, Issue 4

A systematic study of equi-isoclinic n -tuples of the Grassmann manifold G ( d, N ) is initiated. After basic results in the general case, the article focuses on the case d = 2. In particular the lists of all regular equi-isoclinic n -tuples of G (2, 2 n ) and of all equi-isoclinic quadruples of G (2, 6) are established.

The paper deals with rational maps between real algebraic sets. We are interested in the rational maps which extend to continuous maps defined on the entire source space. In particular, we prove that every continuous map between unit spheres is homotopic to a rational map of such a type. We also establish connections with algebraic cycles and vector bundles.

Let Σ be the point-line truncation of an augmented J -Grassmann geometry of a spherical building B . We show that if | J | = 1, or if | J | ≥ 2 and some additional conditions hold, then the convex subspace closure in Σ of the point shadow of an apartment of B is the entire space Σ.

An equifacetal simplex, in which all facets are congruent, has a unique center. The center conjecture states that a simplex that has a unique center must be equifacetal. A strong version of the conjecture is proved in dimensions at most six by showing that there is an explicit list of centers, defined for all simplices, whose coinciding implies the simplex is equifacetal. It remains an open problem whether the conjecture is true in dimensions greater than six.

We construct polarities for arbitrary shift planes and develop criteria for conjugacy under the normalizer of the shift group. Under suitable assumptions (in particular, for finite or compact planes) we construct all shift groups on a given plane, and our constructions yield all conjugacy classes of polarities. We show that a translation plane admits an orthogonal polarity if, and only if, it is a shift plane. The corresponding planes are exactly those that can be coordinatized by commutative semifields. The orthogonal polarities form a single conjugacy class. Finally, we construct examples of compact connected shift planes with more conjugacy classes of polarities than the corresponding classical planes.

Let K be a Lie group, modeled on a locally convex space, and M a finite-dimensional paracompact manifold with corners. We show that each continuous principal K -bundle over M is continuously equivalent to a smooth one and that two smooth principal K -bundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.