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

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

Inhalt

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