On Grassmannizable Group 3-Webs

Abstract

The authors prove that the Lie group G generating a Grassmannizable group 3-web GGW is the group of parameters of the group of similarity transformations of an (r − 1)-dimensional affine space A r−1 . The transitive action of the group G on itself is an r-parameter subgroup B(r) of the group A(r 2 + r) of affine transformations z I = a I J x J + b I , I, J = 1, . . . , r, which is the direct product of the one-dimensional group of homotheties z 1 = kx 1 and r − 1 one-dimensional groups of affine transformationswhere all r groups have the same homothety coefficient k. Conversely, the Lie group B(r) described above generates a Grassmannizable group 3-web GGW. The Lie group G is solvable but not nilpotent. Grassmannizable WebsWe consider the real projective space P r+1 of dimension r + 1 and the set of all straight lines in this space. The latter set is the Grassmannian G(1, r + 1). Its dimension is 2r: dim G(1, r + 1) = 2r. The bundles S(x) of straight lines with centers x ∈ P r+1 are linear submanifolds in G(1, r +1) of dimension r: dim S(x) = r. A smooth hypersurface X ⊂ P r+1 determines a foliation on the Grassmannian G(1, r + 1). The leaves of this foliation are the bundles S(x) of straight lines with centers at points x ∈ X. We consider three smooth