Fundamental theorem of geometry without the surjective assumption (Q2796078)

The authors of this paper believe that some problems are open, and proceed to solve them ``completely.'' The theorems they prove, claiming to have solved these ``problems'' are:NEWLINENEWLINENEWLINENEWLINETheorem 1.1. Suppose that \(f : {\mathbb R}^n \mapsto {\mathbb R}^n\) is a line-to-line injection. Then \(f\) is an affine transformation if and only if \(f\) is non-degenerate.NEWLINENEWLINENEWLINENEWLINETheorem 1.2. Suppose that \(f : \hat{{\mathbb R}}^n \mapsto \hat{{\mathbb R}}^n\) is a circle-to-circle injection. Then \(f\) is a Möbius transformation if and only if \(f\) is non-degenerate.NEWLINENEWLINENEWLINENEWLINETheorem 1.4. Suppose that \(f : {\mathbb D}^n \mapsto {\mathbb D}^n\) is a geodesic-to-geodesic injection. Then it is an isometry or there exists some affine transformation \(g\) on \({\mathbb R}^n\) such that \(g \circ f\) is an isometry if and only if \(f\) is non-degenerate.NEWLINENEWLINENEWLINENEWLINEHere \({\mathbb D}^n\) stands for the open unit ball, and ``geodesic-to-geodesic'' simply means that \(f\) maps collinear points into collinear points. ``Line-to-line injections'' are usually called \textit{injective lineations}, and are injective mappings that map three collinear points into three collinear points.NEWLINENEWLINENEWLINENEWLINEThe problem is that the results of Theorems 1.1, 1.2, and 1.4 were far from open, in fact vastly more ``complete solutions'' have been provided in the literature.NEWLINENEWLINENEWLINENEWLINERegarding Theorem 1.1, a major breakthrough was achieved by \textit{D. S. Carter} and \textit{A. Vogt} [Mem. Am. Math. Soc. 235, (1980; Zbl 0444.51004)], who determined the general form of lineations between both affine and projective planes. Theorem 1.1 can be found, in a vastly more general setting (where the affine spaces need not be over the field of real numbers, and the mappings need not be defined on the entire space), in \textit{M. Saltzwedel}'s [Kennzeichnung von Lineationen desarguesscher affiner Räume. Hamburg: Univ., FB Math. (1995; Zbl 0841.51001); J. Geom. 56, No. 1--2, 142--160 (1996; Zbl 0855.51003); Butzer, P. L. (ed.) et al., Karl der Große und sein Nachwirken. 1200 Jahre Kultur und Wissenschaft in Europa. Band 2: Mathematisches Wissen. Turnhout: Brepols. 287--296 (1998; Zbl 0960.51002)] (see also [\textit{A. Brezuleanu} and \textit{D.-C. Rădulescu}, J. Geom. 23, 45--60 (1984; Zbl 0553.51010)]). Even Theorem 3.6, which characterizes injective lineations defined on convex domains \(D\) in \({\mathbb R}^2\), and which was used in the Proof of Theorem 1.1 for the case \(n=2\) is not new, as it follows from Corollary 3.2 in [\textit{A. Brezuleanu} and \textit{D.-C. Rădulescu}, Abh. Math. Semin. Univ. Hamb. 55, 171--181 (1985; Zbl 0595.51006)].NEWLINENEWLINENEWLINENEWLINEThe case \(n=2\) of Theorem 1.2 was proved in greater generality in formula (8) of Theorem 8 in the reviewer's [Indag. Math., New Ser. 11, No.3, 453--462 (2000; Zbl 0987.51010)]. Anyway, Theorem 1.2 follows quite readily from Theorem 1.1 and results known even to the authors (its proof, on page 6831 taking four lines) .NEWLINENEWLINENEWLINENEWLINEFinally, Theorem 1.4 also follows from Corollary 3.2 in [\textit{A. Brezuleanu} and \textit{D.-C. Rădulescu}, Abh. Math. Semin. Univ. Hamb. 55, 171--181 (1985; Zbl 0595.51006)].NEWLINENEWLINENEWLINENEWLINEAnother reference relevant to this line of thought, which is, as all those cited above, absent from the bibliography of this paper, is \textit{R. Höfer } [J. Geom. 61, No. 1--2, 56--61 (1998; Zbl 0899.51014)].NEWLINENEWLINENEWLINENEWLINEThe authors also spent four bizarre pages, all of subsection 2.2 of this paper, on proving by means of computations statements that the authors claim to be ``generalizations'' of Pappus's theorem. These are, in fact, \textit{specializations}, all of which hold in Desarguesian projective (and thus affine) planes, whose proof by use of variants of Desargues's theorem is immediate and can be found in Theorem 2.6.9 on page 68 of \textit{A. Heyting} [Axiomatic projective geometry. 2nd ed. Groningen: P. Noordhoff N.V. Amsterdam: North- Holland Publishing Company (1980; Zbl 0454.51001)].