On orthogonality-preserving plücker transformations of hyperbolic spaces

Abstract

A complete overview of all orthogonality-preserving Plücker transformations in finite dimensional hyperbolic spaces with dimension other than three is given. In the Cayley-Klein model of such a hyperbolic space all Plücker transformations are induced by collineations of the ambient projective space.

Harmonic mappings and hyperbolic Plücker transformations

“…In [12] there is shown that the structure (G, ∼) forms a Plücker space. Now we investigate the Plücker transformations of this Plücker space in the 3-dimensional case.…”

“…Now we investigate the Plücker transformations of this Plücker space in the 3-dimensional case. Each Q-collineation 1 induces a Plücker transformation of (G, ∼) [12]. But the question is if all Plücker transformations of (G, ∼) are induced by Q-collineations.…”

“…In fact it is sufficient to demand the if part of this characterization (Proposition 2). In [12] the same Plücker transformations are discussed in hyperbolic planes and hyperbolic spaces with dimension ≥ 4. It turns out [12] that those Plücker transformations are exactly the bijections of the set of hyperbolic lines that come from collineations of the hyperbolic space.…”

“…In [12] the same Plücker transformations are discussed in hyperbolic planes and hyperbolic spaces with dimension ≥ 4. It turns out [12] that those Plücker transformations are exactly the bijections of the set of hyperbolic lines that come from collineations of the hyperbolic space. In the Cayley-Klein model, which is based upon an absolute quadric Q, such a "hyperbolic collineation" extends to a unique collineation of the ambient projective space.…”

Coplanarity of lines in projective and polar Grassmann spaces

“…In metric geometry lines intersecting at right angles play an essential role. It has been proved in [8,11] for elliptic spaces, in [12] for symplectic spaces, and in [19][20][21][22]24] for hyperbolic spaces that transformations which preserve ortho-adjacency of lines are induced by collineations that preserve orthogonality, unless the underlying projective space has three dimensions. These results were generalized for k-subspaces in metric-projective settings in [28].…”