Manifolds, particularly space curves: basic notions 1 The first groundform, the covariant metric tensor 11 The second groundform, Meusnier's theorem 19 Transformation groups in the plane 28 Co- and contravariant components for a special affine transformation in the plane 29 Surface vectors 32 Elements of tensor calculus 36 Generalization of the first groundform to the space 46 The covariant (absolute) derivation 57 Examples from elasticity theory 61 Geodesic lines 63 Main curvature, average and Gaussian curvature 67 Spherical image of a surface and third groundform 74 Parallel transport according to Levi-Civita: integrability conditions 76 Equations of Mainardi-Codazzi and Gauss 80 Remarks on non Euklidian geometries and special relativity theory 82 General relativity theory 88 Gravitational lensing and redshift 98 Exercices 102
