The recent physical interpretation of intrinsic differential geometry
of spaces has stimulated the study of this subject. Riemann pro-
proposed the generalization, to spaces of any order, of the theory of
surfaces, as developed by Gauss, and introduced certain fundamental
ideas in this general theory. From time to time important con-
contributions to the theory were made by Bianchi, Beltrami, Christoffel,
• Schur, Voss and others, and Ricci coordinated and extended the
theory with the use of tensor analysis and his Absolute Calculus.
Recently there has been an extensive study and development of
Riemannian Geometry, and this book aims to present the existing
theory.
Throughout the book constant use is made of the methods of
tensor analysis and the Absolute Calculus of Ricci and Levi-Civita.
The first chapter contains an exposition of tensor analysis in form
and extent sufficient for the reader of the book who has not
previously studied this subject. However, it is not intended that
the exposition shall give an exhaustive foundational treatment of
the subject.
Most, if not all, of the contributors to the theory of Riemannian
Geometry have limited their investigations to spaces with a metric
.-^defined by a positive definite quadratic differential form. How-
However, the theory of relativity deals with spaces with an indefinite
fundamental form. Consequently the former restriction is not made
in this book. Although many results of the older theory have
been modified accordingly, much remains to be done in this field.
<* The theory of parallelism of vectors in a general Riemannian