Differential geometry of curves and surfaces : a concise guide
著者
書誌事項
Differential geometry of curves and surfaces : a concise guide
Birkhäuser, c2006
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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
目次
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
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