Die innere Geometrie der metrischen Räume

The inner geometry of a surface is the study of those properties that remain unchanged under isometric mappings, thus depending only on its first fundamental form. It was established by C. F. Gauss through the discovery that the product of the principal curvatures of a surface is an isometric invariant. B. Riemann extended this theory in his habilitation thesis to multidimensional and thus also to abstract manifolds. Initially, only the study of such manifolds was considered, where the arc element is given by the square root of a quadratic differential form. P. Finsler developed the inner geometry in his dissertation based on a general arc element, a possibility that had already been recognized by B. Riemann. Since the classical investigations by J. Hadamard on surfaces of constant negative curvature and by D. Hilbert on the existence of extremals in variational problems, the understanding has increasingly emerged that a large part of the methods, particularly those developed in global differential geometry, require only the topological and metric structure of the manifolds, not their differentiability structure. The concept of metric space created by Fréchet made it possible to establish inner geometry on a basis free from differentiability assumptions. However, initially, the topology of metric spaces was the focus of interest. It was only with K. Menger that a systematic study of isometric invariants began. Meanwhile, an extensive literature has emerged. The main results are found in the three books by A. D. Alexandrow, L. M. Blumenthal, and H.