**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

**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 ("

**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*(

**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*,

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