For a number of years, beginning with the early 70's, the authors have been
delivering lectures On the fundamentals of geometry and topology in the Faculty of
Mechanics and Mathematics of Moscow State University. This text·book is the result
of this work. We shall recall that for a long period of time the basic elements of
modem geometry and topology were not included, even by departments and faculties
ofmathematics, as compulsory subjects in a university-level mathematical education.
The standard courses in classical differential geometry have gradually become
outdated, and there has been, hitherto, no unanimous standpoint as to which parts of
modem geometry should be viewed as abolutely essential to a modem mathematical
education. In view of the necessity of using a large number of geometric concepts
and methods, a modernized courSe in geometry was begun in 1971 in the Mechanics
division of the Faculty ofMechanics and Mathematics ofMoscow State University.
In addition to the traditional geometry of curves and surfaces, the course included the
fundamental priniciples of tensor analysis, Riemannian geometry and topology.
Some time later this course was also introduced in the division of mathematics. On
the basis ofthese lecture courses, the following text-books appeared:
S.P. Novikov: Differential Geometry, Pans I and II, Research Institute of
Mechanics of Moscow State University, 1972.
S.P. Novikov and A.T. Fomenko: Differential Geometry, Pan III, Research
Institute of Mechanics ofMoscow State University, 1974.
The present book is the outcome of a revision and updating of the
above-mentioned lecture noteS. The book is intended for the mathematical, physical
and mechanical education of second and third year university students. The
minimum abstraciedness of the language and style of presentation" of the milterial,
consistency with the language of mechanics and physics, and the preference for the
material imponant for natural sciences were the basic principles ofthe presentation.
At the end of the book are several Appendices which may serve to diversify
the material presented in the main text So, for the purposes of mechanical and
physical education the information On elementary groups of transformations and
geometric elements of variational calculus can be extended using these Appendices.
For mathematicians, the Appendices may serve to enrich their knowledge of
Lobachevsky geometry and homology theory. We believe that Appendices 2 and 3
are very instructive for those who wish to become acquainted with the simplest
geometric ideas fundamental to physics. Appendix 7 includes selected problems and
exercises for the course.
The list ofreferences may assist in further independent study. A more detailed
text-book which provides deeper insight into geometry and its applications is Modern
Geometry [1].