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.

Let Π h be a 3-dimensional hyperbolic space with Euclidean ground field K. There is a certain procedure by which any harmonic mapping of the projective line over the unique quadratic extension of K induces an orthogonality-preserving Plücker transformation of Π h and, conversely, any orthogonality-preserving Plücker transformation of Π h is induced by such a harmonic mapping.

We show that the automorphisms of the flag space associated with a 3-dimensional projective space can be characterized as bijections preserving a certain binary relation on the set of flags in both directions. From this we derive that there are no other automorphisms of the flag space than those coming from collineations and dualities of the underlying projective space. Further, for a commutative ground field, we discuss the corresponding flag variety and characterize its group of automorphic collineations. Mathematics Subject Classification (2000): 51M35, 51N15, 15A75, 14M15.

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