The original Russian edition of this book is the fifth in my series
"Lectures on Geometry. " Therefore, to make the presentation relatively
independent and self-contained in the English translation, I have added
supplementary chapters in a special addendum (Chaps. 3Q-36), in which
the necessary facts from manifold theory and vector bundle theory are
briefly summarized without proofs as a rule. In the original edition,
the book is divided not into chapters but into lec- tures. This is
explained by its origin as classroom lectures that I gave. The principal
distinction between chapters and lectures is that the material of each
chapter should be complete to a certain extent and the length of
chapters can differ, while, in contrast, all lectures should be
approximately the same in length and the topic of any lecture can change
suddenly in the middle. For the series "Encyclopedia of Mathematical
Sciences," the origin of a book has no significance, and the name
"chapter" is more usual. Therefore, the name of subdivisions was changed
in the translation, although no structural surgery was performed. I have
also added a brief bibliography, which was absent in the original
edition. The first ten chapters are devoted to the geometry of affine
connection spaces. In the first chapter, I present the main properties
of geodesics in these spaces. Chapter 2 is devoted to the formalism of
covariant derivatives, torsion tensor, and curvature tensor. The major
part of Chap.