A course in metric geometry.

*(English)*Zbl 0981.51016
Graduate Studies in Mathematics. 33. Providence, RI: American Mathematical Society (AMS). xiv, 415 p. (2001).

This textbook is a graduate course on metric geometry. It does not only provide the concept of metric spaces but it delves deeper into the realm of metrics. The authors claim to open a vast panorama of ‘geometrical mathematics from a metric viewpoint’ and in fact the notion works out. To mention but a few, they cover areas like ‘Spaces of Bounded Curvature’, ‘Smooth Length Structures’, the ‘Curvature of Riemannian Metrics’ as well as ‘Large-scale Geometry’. Scarce as the figures may be the book is still a graphic and colourful guide into quite some parts of geometry based on a length structure viewpoint. There are also helpful exercises for the reader in plentiful supply. An interesting aspect about the book is that many notions essentially belong to metric geometry though their definition is based on differential geometric concepts. This revelation also gives way to a more graphic access to many fields of geometry.

Reviewer: Johann Lang (Graz)