Differential Geometry of Curves and Surfaces

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.

Central topics covered include curves, surfaces, geodesics, intrinsic geometry, and the Alexandrov global angle comparision theorem Many nontrivial and original problems (some with hints and solutions) Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels

Autorentext
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.


Klappentext
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.

Zusammenfassung
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)

Inhalt
Theory of Curves in Three-dimensional Euclidean Space and in the Plane.- Extrinsic Geometry of Surfaces in Three-dimensional Euclidean Space.- Intrinsic Geometry of Surfaces.

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