Grassmannian structures on manifolds

Semiintegrable almost Grassmann structures

“…are the curvature 2-forms of this structure. Note that Dhooghe in [4] and [5] gave the structure equations of an almost Grassmann structure in the form close to our equations (1.4). His equations differ from equations (1.4) only by the additional term Ω * in the right-hand side of the fourth equation.…”

On the Theory of Almost Grassmann Structures

“…Note that in the main parts of [12] and [13] the author considered torsionfree almost Grassmann structures. He did not have results of our Theorems 4.2, 4.3 and 4.4 and erroneously assumed that for p > 2, q > 2 there exist torsion-free almost Grassmann structures that are not locally Grassmann (locally flat) structures.…”

“…Bailey and Eastwood [9] extended the theory of local twistors, which was known for four-dimensional conformal structures, to the almost Grassmann structures (they called them the paraconformal structures). Dhooghe ([12], [13]) considered almost Grassmann structures (he called them Grassmannian structures) as subbundles of the second order frame bundle and constructed a canonical normal connection for these structures. The structure equations derived for the Grassmann structures in [13] are very close to the structure equations of the Grassmann structures considered in the present paper.…”

“…1. We consider a projective model of a Grassmann manifold while in the papers [12] and [13] (and in most of other papers on this subject) a vector model was considered. This is the reason that in our presentation both the dimension n of the original projective space and the dimension m of the generating element of the Grassmann manifold G(m, n) are one unit less than in those papers.…”

“…In the main parts of [12] and [13] if p > 2 and q > 2, then torsion-free almost Grassmann structures are locally Grassmann structures. This is the reason that the results of the main parts of [12] and [13] valid only for p = 2, q > 2; p > 2, q = 2, and p = q = 2.…”