The simplest polymetric geometries. I
In the theory of physical structures, the term ``distance'' is used for denoting an arbitrary real-valued function defined on the set of all ordered pairs of points. This function need not satisfy the usual axioms of distance, but there must exist a number \(k\) such that the set of all mutual distances between any \(k\) points satisfies a certain functional equation. (Recall, for example, that 6 distances between any 4 points in the Euclidean space vanish the Cayley--Menger determinant. Hence, we can consider the Euclidean 3-space as a ``physical structure of range 4''). In his previous paper [Sib. Math. J. 25, 764-774 (1984); translation from Sib. Mat. Zh. 25, No. 5(147), 99-113 (1984; Zbl 0562.51012)], the author proved that such a ``distance'' admits a nontrivial local Lie group of transformations which preserve the distance between any two points. Moreover, he succeeds in using these Lie groups for classifying ``physical structures'' under some natural conditions. The article under review is the first part of the author's investigations, where he introduces a new object called a ``polymetric physical structure'' which is a manifold \(\mathfrak M\) endowed with a finite set of ``distances'' \(f^i: \mathfrak M \to \mathfrak M\), \(i=1,2,\dots ,s\). The aim of the author is to generalize his results mentioned above to this new object. He constructs a local Lie group associated with the ``polymetric physical structure'' and calculates the dimension of the group. It is announced that this information will be used in the second part of the article and will allow us to classify ``polymetric physical structures'' of range 2 and 3. The proofs are based on the techniques of classical local Lie groups.
scientific article; zbMATH DE number 1431644