Dedicated to Adriano Barlotti on the occasion of his 80th birthday, in friendship
Let A" be a real inner product space of finite or infinite dimension ^2, and let Ο Ο Ξ be a fixed real number. The following results will be presented in this note.
A. A surjective mapping Ο : X -Β» X preserving Lorentz-Minkowski distances 0 and Q in one direction must be a Lorentz transformation.
B. The causal automorphisms of X, dim X ^ 3, are exactly the products Ξ΄ Ξ», where Ξ» is an orthochronous Lorentz transformation and Ξ΄ a dilatation jc -> Ξ±Ο, R 9 a > 0.
C. If Q > 0, there exist A" and an injective Ο : X -> X preserving Lorentz-Minkowski distance Ο, such that Ο is not a Lorentz transformation. This result can be extended, mutatis mutandis, to Euclidean and Hyperbolic Geometry.
If X is finite-dimensional, result A is an immediate consequence of the following theorem of Benz-Lester ([4], [12], [13], [5]).
Theorem 1. Suppose that X is a real inner product space of finite dimension ^2 and that Ο Ο 0 is a fixed real number. Ifa:X-*X satisfies for all x, y Ξ΅ X, where /(x, y) designates the Lorentz-Minkowski distance ofx, y, then Ο must be a Lorentz transformation.
It could be possible that Theorem 1 also holds true in the infinite-dimensional case provided that Ο < 0. However, a proof, if it exists, is not yet known. Result C shows that Theorem 1 cannot be extended to the infinite-dimensional case if Ο > 0, not even in the injective case.