28377 - Geometry 3

Learning outcomes

The aim of the course is to give the student a solid understanding of the most important features of the local theory of curves and surfaces in euclidean three-dimensional space. This will enable the student to go on to learn more advanced topics in differential geometry.

Course contents

A reminder on the structure of the group of isometries of the plane and of three-dimensional space. Parameterization of the orthogonal group via unit length quaternions. Parameterized curves, equivalence, velocity vector of a parameterized curve. Length of a path, arc length parameterization.
Curvature of a plane curve, its meaning and how to compute it. Frenet theory for curves in three-dimensional space: Frenet frame, curvature torsion, fundamental equations. Existence and unicity, up to isometries, of curves with given curvature and torsion. Examples of some remarkable curves.
Theory of surfaces in three-dimensional space. Parametric and implicit equations.   Tangent space and Gauss map of an oriented surface. First and second fundamental form. Curvature, principal directions, mean and Gaussian curvature, and how to compute them. Types of points on a surface, normal curvature. Meusnier and Euler theorem. The gaussian curvature is an isometry invariant. Some remakable class of surfaces: ruled surfaces and surfaces of revolution. Introduction to geodetic curves on a surface, and the Gauss Bonnet theorem for a geodesic triangle.  Some notions of spheric and hyperbolic geometry.

Teaching methods

Lectures

Assessment methods

Oral examination

Office hours

See the website of Luca Migliorini