Differential Geometry of Curves and Surfaces: A Concise Guide

ISBN-13: 9780817643843 / Angielski / Miękka / 2005 / 224 str.

Victor A. Toponogov; Vladimir Y. Rovenski

Opis

This concise guide to the differential geometry of curves and surfaces can be recommended to ?rst-year graduate students, strong senior students, and students specializing in geometry. The material is given in two parallel streams. The ?rst stream contains the standard theoretical material on differential ge- etry of curves and surfaces. It contains a small number of exercises and simple problems of a local nature. It includes the whole of Chapter 1 except for the pr- lems (Sections 1.5, 1.7, 1.10) and Section 1.11, about the phase length of a curve, and the whole of Chapter 2 except for Section 2.6, about classes of surfaces, T- orems 2.8.1 2.8.4, the problems (Sections 2.7.4, 2.8.3) and the appendix (S- tion 2.9). The second stream contains more dif?cult and additional material and for- lations of some complicated but important theorems, for example, a proof of A.D. Aleksandrov s comparison theorem about the angles of a triangle on a convex 1 surface, formulations of A.V. Pogorelov s theorem about rigidity of convex s- faces, and S.N. Bernstein s theorem about saddle surfaces. In the last case, the formulations are discussed in detail. A distinctive feature of the book is a large collection (80 to 90) ofnonstandard andoriginalproblems that introduce the student into the real world of geometry."

Recenzje

From the reviews:

"This book by the late author covers ... the subjects which are normally taught in a course on the differential geometry of curves and surfaces. ... It can be recommended for first-year graduate students and also for use in the classroom. ... the book is rich in geometry and concrete examples. ... the book is very welcome since it is an original contribution in various aspects and gives a number of geometric insights ... . Numerous illustrations make the reading enjoyable." (Wolfgang Kühnel, Mathematical Reviews, Issue 2006 m)

"Toponogov's 'concise guide' to elementary differential geometry has the potential to be a useful reference and/or review book ... ."(Fernando Q. Gouvèa, MathDL, March, 2006)

"This book, written by one of the Russian masters, offers a comprehensive introduction to classical differential geometry of curves and surfaces. Only a basic background in analysis and linear algebra is needed to follow the presentation. ... A distinctive feature of the book is that, apart from a large number of exercises, it also contains many original problems with outlines of solutions." (A. Cap, Monatshefte für Mathematik, Vol. 155 (1), September, 2008)

Spis treści

Chapter 1 Curves in a 3-dimensional Euclidean space and in the plane: Preliminaries.- Definition and methods of curves presentation.- Tangent line and an osculating plane.- Length of a curve.- Problems: plane convex curves.- Curvature of a curve.- Problems: curvature of plance curves.- Torsion of a curve.- Frenet formulas and the natural equation of a curve.- Problems: space curves- Phase length of a curve and Fenchel-Reshetnyak inequality.- Exercises Chapter 2 Extrinsic geometry of surfaces in a 3-dimensional Euclidean space.- Definition and methods of generating surfaces.- Tangent plane.- First fundamental form of a surface.- Second fundamental form of a surface.- The third fundamental form of a surface.- Classes of surfaces.- Some classes of curves on a surface.- The main equations of the surfaces theory.- Appendix: Indicatrix of a surface of revolution.- Exercises Chapter 3 Intrinsic geometry of surfaces.- Introducing notions.-Covariant derivative of a vector field.- Parallel translation of a vector along a curve on a surface.- Geodesics.- Shortest paths and geodesics.- Special coordinate system.- Gauss-Bonet theorem and comparison theorem for the angles of a triangle.- Local comparison theorems for triangle.- Alexandrov comparison theorem for the angles of a triangle.- Problems.- Bibliography.- Index

O autorze

Differential Geometry of Curves and Surfaces: A Concise Guide presents traditional material in classicial differential geometry field along with important ideas of Riemannian geometry. The reader is introduced to curves, then to surfaces, and finally to more complex topics. Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels.
The work covers central concepts including curves, surfaces, geodesics, intrinsic geometry, the Aleksandrov global angle comparison theorem, and the celebrated Toponogov Comparison Theorem . This rigorous exposition, with well-motivated topics and many nontrivial and original problems (some with hints and solutions), is ideal for advanced undergraduate and first-year graduate students seeking to enter the fascinating world of geometry.

Blurb

The study of curves and surfaces forms an important part of classical differential geometry. Differential Geometry of Curves and Surfaces: A Concise Guide presents traditional material in this field along with important ideas of Riemannian geometry. The reader is introduced to curves, then to surfaces, and finally to more complex topics. Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels.

Key topics and features:

* Covers central concepts including curves, surfaces, geodesics, and intrinsic geometry

* Substantive material on the Aleksandrov global angle comparison theorem, which the author generalized for Riemannian manifolds (a result now known as the celebrated Toponogov Comparison Theorem, one of the cornerstones of modern Riemannian geometry)

* Contains many nontrivial and original problems, some with hints and solutions

This rigorous exposition, with well-motivated topics, is ideal for advanced undergraduate and first-year graduate students seeking to enter the fascinating world of geometry.