包邮微分几何基础-第2版-(影印版)

PrefaceContents1．Curves in the plane and in space1．1 What is a curve 1．2 Arc-length1．3 Reparametrization1．4 Closed curves1．5 Level curves versus parametrized curves2．How much does a curve curve 2．1 Curvature2．2 Plane curves2．3 Space curves3．Global properties of curves3．1 Simple closed curves3．2 The isoperimetric inequality3．3 The four vertex theorem4．Surfaces in three dimensions4．1 What is a surface 4．2 Smooth surfaces4．3 Smooth maps4．4 Tangents and derivatives4．5 Normals and orientability5．Examples of surfaces5．1 Level surfaces5．2 Quadric surfaces5．3 Ruled surfaces and surfaces of revolution5．4 Compact surfaces5．5 Triply orthogonal systems5．6 Applications of the inverse function theorem6．The flrst fundamental form6．1 Lengths of curves on surfaces6．2 Isometries of surfaces6．3 Conformal mappings of surfaces6．4 Equiareal maps and a theorem of Archimedes6．5 Sphericalgeometry7．Curvature of 8urfaces7．1 The second fundamental form7．2 The Gauss and Weingarten maps7．3 Normal and geodesic curvatures7．4 Parallel transport and covariant derivative8．Gaussian， mean and principal curvatures8．1 Gaussian and mean curvatures8．2 Principal curvatures of a surface8．3 Surfaces of constant Gaussian curvature8．4 Flat surfaces8．5 Surfaces of constant mean curvature8．6 Gaussian curvature of compact surfaces9．Geodesics9．1 Definition and basic properties9．2 Geodesic equations9．3 Geodesics on surfaces of revolution9．4 Geodesics as shortest paths9．5 Geodesic coordinates10．Gauss' Theorema Egregium10．1 The Gauss and Codazzi-Mainardi equations10．2 Gauss' remarkable theorem10．3 Surfaces of constant Gaussian curvature10．4 Geodesic mappings11．Hyperbolic geometry11．1 Upper half-plane model11．2 Isometries of H11．3 Poincare disc model11．4 Hyperbolic parallels11．5 Beltrami-Klein model12．Minmal surfaces12．1 Plateau's problem12．2 Examples of minimal surfaces12．3 Gauss map of a minimal surface12．4 Conformal parametrization of minimal surfaces12．5 Minimal surfaces and holomorphic functions13．The Gauss-Bonnet theorem13．1 Gauss-Bonnet for simple closed curves13．2 Gauss-Bonnet for curvilinear polygons13．3 Integration on compact surfaces13．4 Gauss-Bonnet for compact surfaces13．5 Map colouring13．6 Holonomy and Gaussian curvature13．7 Singularities of vector fields13．8 Critical pointsA0．Inner product spaces and self-adjoint linear mapsA1．Isometries of Euclidean spacesA2．Mobius transformationsHints to selected exercisesSolutionsIndex