Course information

Curves: General concepts, parametric curves in space, implicit form. Arc length, curvature, torsion, Frenet-Serret formulas, spherical indicatrices. Local form of curve. Osculating circles and spheres, involutes and evolutes. Fundamental Theorem of curves. Plane curves, envelopes, convexity. Closed curves and global theorems: Jordan Curve Theorem, Schonflies Theorem, Riemann-Hopf Theorem, Four Vertex Theorem, Isoperimetric Inequality. Surfaces: General concepts, definition of a surface (surface patch, simple surface, smooth surface), tangent space, atlases, orientability. First and second fundamental forms. Gauss-Rodrigues map, Weingarten map, shape operator. Normal curvature, principal directions, Gaussian curvature, mean curvature, geodesics, lines of curvature, asymptotic curves. Classification of surface points, local form of a surface. Theorema Egregium, Gauss-Bonnet Theorem. Surface mappings: isometric, conformal, equiareal, minimal surfaces.