Preface

Text-book on Differential Geometry Geometry of curves and surfaces is intended for first level University students and for those students who are specializing in geometry. The main features of this manuscript are the following.

1. The learning material is given in two parallel streams.

The first stream contains the standard theoretical material, according to the standard University program, on  Differential Geometry. It contains a small number of standard exercises and simple problems of local nature. In accordance with the author, it includes the whole of Chapter 1 except for the problems and the whole of Chapter 2 except for Section 2.6 about classes of surfaces, Theorems 2.8.1-2.8.4  and the problems.

The second stream contains more difficult and additional material and formulations of some complicated, but important theorems. For example, proof of A.D. Alexandrov's comparison theorem about the angles of a triangle on a convex surface, formulations of A.V. Pogorelov's theorem about rigidity of convex surfaces and S.N. Bernstein's theorem about saddle surfaces. In the last case the formulations are discussed in detailed form.

2. The text-book contains a large number (80 to 90) of non-standard and original problems. Most of them are new and could not be found in other training publications (text-books or books of problems) on Differential Geometry. The solutions of these problems require an inventiveness and geometrical intuition on the part of the students. In this respect, the text-book is not far from W. Blaschke's well-known manuscript ("Elementary Differential Geometry...", Berlin, 1930), but contains a lot of Problems, more contemporary in theme. The key of these problems is the notion of a curvature: the curvature of a curve, principal curvatures and Gaussian curvature of a surface. Almost all problems are given with their solutions, although the hope of the author is that an honest student will solve them without assistance, and only in exceptional cases, will look at the text for a solution. Since the problems are given in an increasing order of difficulty, even most difficult of them is available to a reader. In some cases only short instructions are given. In the author's opinion, namely, a selection of a number of original problems makes this text-book interesting and useful.

3. Chapter 3 "Intrinsic geometry of a surface" starts from the main notion of a covariant derivative of a vector field along a curve. The definition is based on extrinsic geometrical properties of a surface. Then it is proven that the covariant derivative of a vector field is an object of intrinsic geometry of a surface, and the later training material is not related with an extrinsic geometry. So, Chapter 3 can be considered as an introduction to n-dimensional Riemannian geometry, that keeps the simplicity and clearness of  2-dimensional case.
The main theorems about geodesics and shortest paths are proven by the methods that can be easily extended to n-dimensional situations almost without alterations. Alexandrov comparison Theorem for the angles of a triangle is the highest point in Chapter 3.

The author is one of the founders of the CAT(k)-spaces theory (the initials are in honor of E.Cartan, A.D.Alexandrov and V.A.Toponogov), where the comparison theorem for the angles of a triangle, or more exactly, its generalization by author for multi-dimensional Riemannian manifolds, takes the part of the basic property of  CAT(k)-spaces.

Acknowledgments.
Seizing the opportunity, the author greatly thanks his learners and colleagues, who have contributed to this volume. The essential help was given by E.D. Rodionov, V.V. Slavski, V.Yu. Rovenski, V.V. Ivanov, V.A. Sharafutdinov, V.K. Ionin.

Prof. Victor A. Toponogov

Chief Scientist
Department of Analysis and Geometry
Sobolev Institute of Mathematics
Siberian Branch of the Russian Academy of Sciences
Novosibirsk-90, 630090, Russia