This is the final volume of a three volume
collection devoted to the geometry, topology, and curvature of
2-dimensional spaces. The collection provides a guided tour through a
wide range of topics by one of the twentieth century's masters of
geometric topology. The books are accessible to college and graduate
students and provide perspective and insight to mathematicians at all
levels who are interested in geometry and topology.
Einstein showed how to interpret gravity as the dynamic response to
the curvature of space-time. Bill Thurston showed us that
non-Euclidean geometries and curvature are essential to the
understanding of low-dimensional spaces. This third and final volume
aims to give the reader a firm intuitive understanding of these
concepts in dimension 2. The volume first demonstrates a number of the
most important properties of non-Euclidean geometry by means of simple
infinite graphs that approximate that geometry. This is followed by a
long chapter taken from lectures the author gave at MSRI, which
explains a more classical view of hyperbolic non-Euclidean geometry in
all dimensions. Finally, the author explains a natural intrinsic
obstruction to flattening a triangulated polyhedral surface into the
plane without distorting the constituent triangles. That obstruction
extends intrinsically to smooth surfaces by approximation and is
called curvature. Gauss's original definition of curvature is
extrinsic rather than intrinsic. The final two chapters show that the
book's intrinsic definition is equivalent to Gauss's extrinsic
definition (Gauss's “Theorema Egregium” (“Great
Theorem”)).
Readership
Graduate and undergraduate students and researchers
interested in topology.