27013 - Geometry of Curves and Surfaces
The goal of the subject "Geometry of curves and surfaces" is the study of the differential geometry of curves and surfaces in the euclidean plane and space. Class syllabus:
Part 1. Regular plane curves. Frénet's frame, tangent and normal vector fields along a curve, curvature, arc length. Fundamental Theorem for plane curves.
Part 2. Biregular spatial curves, Frénet frame (tangent, normal and binormal fields), arc length, torsion, curvature, evolute. Fundamental Theorem for spatial curves. Local canonical form.
Part 3. Regular surfaces. Local theory: 2-function graphs, charts and regular values of 3-functions. Examples. Parametrized surfaces. Curves in surfaces and Tangent plane.
Charts, coordinate vector fields, differentiable functions and maps. First fundamental form: lengths, angles and areas.
Part 4. Geometry of Surfaces. Geodesic and normal curvature. Second fundamental form and Gauss map. Types of points, principal, normal and Gauss curvature. Principal directions, asympotic curves, umbilic points. Vector and direction fields.
Ruled and minimal surfaces.
Part 5. Intrinsic Geometry. Covariant derivative and Gauss Theorema Eggregium. Isometries, conformal maps and isothermal coordinates. Geodesics and exponential map: distance and convexity. Gauss-Bonnet Theorems.
Some other topics, as those related with global geometry of curves and surfaces will be developed by the students in groups: Isoperimetric inequality, Four-vertex Theorem, Regular neighbourhoods of compact curves and surfaces, Differentiable Jordan Curve Theorem, Fenchel's Theorem, hyperbolic geometry, minimal and ruled surfaces,etc.