Learning outcomes of the course unit
From this course the student will learn how to:
- analyse the fundamental geometric properties of differentiable curves and surfaces in the 3D space;
- distinguish between surfaces up to isometries;
- understand the key logic steps of the proofs;
- express rigorously the learned notions.
The course needs notions of linear algebra, topology and analysis, that is the arguments of Geometria 1a-1b, Geometria 2a, Analisi 1a-1b.
Course contents summary
Geometry of curves and surfaces in the 3D space.
Conics and quadrics: definition, examples and classification.
Differentiable curves in 3D space: definition and examples, curve's length, parametrization, regularity, Frenet's Formula, torsion and curvature, Fundamental theorem of curve's local theory.
Regular surfaces: definition, surfaces preimage of a regular value, surfaces of revolution and ruled surfaces. Differentiable function and tangent space, normal vector field, orientability. Gauss' map, first and second fundamental forms, curvatures. Hilbert's Theorem. Minimal surfaces. Isometries, Theorema Egregium, geodesics. Gauss-Bonnet local theorem.
 M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Dover Publications, 2016.
 M. Abate, F. Tovena, Curve e Superfici, Unitext, Springer, Milano, 2016.
 M. Abate, C. de Fabritiis, Geometria analitica con elementi di algebra lineare, McGraw-Hill Education, 2015.
Lectures and classroom exercises.
During lectures in traditional mode, the
topics will be formally and rigorously presented. The course will give particular emphasis to application and computation aspects, while not letting out
the theoretical aspect. The classroom exercises are aimed at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.
Assessment methods and criteria
The exams consists in a written test (three exercises to be completed in two and a half hours) and an oral examination which will take place in a different date.