Differential Geometry and Its Visualization

1. Curves in Threeâ"dimensional Euclidean Space. 1.1. Points and Vectors. 1.2. Vectorâ"valued Functions of a Real Variable. 1.3. The General Concept of Curves. 1.4. Some Examples of Planar Curves. 1.5. The Arc Length of a Curve. 1.6. The Vectors of the Trihedron of a Curve. 1.7. Frenetâs Formulae. 1.8. The Geometric Significance of Curvature and Torsion. 1.9. Osculating Circles and Spheres. 1.10. Involutes and Evolutes. 1.11. The Fundamental Theorem of Curves. 1.12. Lines of Constant Slope. 1.13. Spherical Images of a Curve. 2. Surfaces in Threeâ"dimensional Euclidean Space. 2.1. Surfaces and Curves on Surfaces. 2.2. The Tangent Planes and Normal Vectors of a Surface. 2.3. The Arc Length, Angles and Gaussâs First Fundamental Coefficients. 2.4. the Curvature of Curves on Surfaces, Geodesic and Normal Curvature. 2.5. The Normal, Principal, Gaussian and Mean Curvature. 2.6. The Shape of a Surface in the Neighbourhood of a Point. 2.7. Dupinâs Indicatrix. 2.8. Lines of Curvature and Asymptotic Lines. 2.9. Triple Orthogonal Systems. 2.10. the Weingarten Equations. 3. The Intrinsic Geometry of Surfaces. 3.1. the Christoffel Symbols. 3.2. Geodesic Lines. 3.3. Geodesic Lines on Surfaces with Orthogonal Parameters. 3.4. Geodesic Lines on Surfaces of Revolution. 3.5. the Minimum Property of Geodesic Lines. 3.6. Orthogonal and Geodesic Parameters. 3.7. Leviâ"civit¡ Parallelism. 3.8. Theorema Egregium. 3.9. Maps Between Surfaces. 3.10. the Gaussâ"bonnet Theorem. 3.11. Minimal Surfaces. 4. Tensor Algebra and Riemannian Geometry. 4.1. Differentiable Manifolds. 4.2. Transformation of Bases. 4.3. Linear Functionals and Dual Spaces. 4.4. Tensors of Second Order. 4.5. Symmetric Bilinear Forms and Inner Products. 4.6. Tensors of Arbitary Order. 4.7. Symmetric and Antiâ"symmetric Tensors. 4.8. Riemann Spaces. 4.9. the Christoffel Symbols. 5. Tensor Analysis. 5.1. Covariant Differentiation. 5.2. the Covariant Derivative of an (R, S)â"tensor. 5.3. the Interchange of Order for Covariant Differentiation and Ricciâs Identity. 5.4. Bianchiâs Identities for the Covariant Derivative of the Tensors of Curvature. 5.5. Beltramiâs Differentiators. 5.6. a Geometric Meaning of the Covariant Differentiation, the Leviâ"civit¡ Parallelism. 5.7. The Fundamental Theorem for Surfaces. 5.8. A Geometric Meaning of the Riemann Tensor of Curvature. 5.9. Spaces With Vanishing Tensor of Curvature. 5.10. An Extension of Frenetâs Formulae. 5.11. Riemann Normal Coordinates and the Curvature of Spaces.