Zur Kennzeichnung distanztreuer Abbildungen in nichteuklidischen Räumen

Abstract

Let (V,K,Q) be a noneuclidean regular metric vector space, ρ a fixed element of K and ϕ: V → V a bijection such that

$$Q(x - y) = \rho \leftrightarrow Q(x^\phi - y^\phi ) = \rho \forall x,y \in v.$$

.

Then, under certain conditions, there exist a semilinear bijection (σ,τ): (V,K) → (V,K) and an element α ∈ K \{O} such that

$$x^\phi = x^\sigma + \mathop 0\limits^{\phi } \wedge Q(x^\sigma ) = \alpha \cdot (Q(x))^\tau \forall x \in v,$$

and in the case ρ ≠ O we get$\alpha = \frac{\rho }{{\rho ^\tau }}$.

Results of this type are stated in [1],[4] – [12],[17] and [18]. Generalizing these results, in this paper it is proved, that the above statement holds in the case 3 ≤ dim V ≤ ∞ ∧ char K arbitrary ∧ ¦K¦ ≥ 4, if ρ is spacelike, and that it is also true in the case 3 ≤ dim V ≤ ∞ ∧ char K ≠ 2,3,5, if ρ is timelike.