Kennzeichnungen hyperbolischer Bewegungen durch Lineationen

A common characterization of classical and relativistic addition

“…If dim X < ∞, Theorem 2 of R. Höfer [3] implies that ϕ must be a hyperbolic motion of V . Höfer's theorem is based on the standard scalar product.…”

“…Hence ψ (x) ∈ ∩ V and thus ψ ( ) ⊆ ∩ V . Now, again, we apply Höfer's Theorem 2, [3], and we get that ψ, restricted on , is a hyperbolic motion. Hence…”

Mappings preserving the area equality of hyperbolic triangles are motions

“…Such characterizations have been shown in a series of papers [12][13][14][15][16][17][18][19][21][22][23] to be equivalent to definability statements which state that certain geometric notions are definable, with certain syntactic constraints on the definiens, in terms of another geometric notion. Characterizations of self-mappings of hyperbolic spaces, going beyond those that have been collected in [2][3][4], can be found in [7][8][9][10][11]13,[15][16][17]21].…”

“…Since α 2 is the product of six point reflections, and each point reflection can be written as a product of two line reflections, α 2 is the product of twelve, i.e., an even number of line reflections, and as such, cannot be a line reflection, so α 2 must be the identity. Since α itself could not have been the identity by [1,§3,8,Satz 23], it follows that α is involutory, and thus, by [1,§3,7,Satz 17], it must be a point reflection or a line reflection. Being a product of an even number of line reflections, α cannot be a line reflection, so it must be a point reflection.…”