Learning objectives
From this course the student will learn how to:
- analyze geometric properties of differentiable curves and surfaces in the 3D space;
- understand the key logic steps of the proofs;
- express rigorously the learned notions.
Prerequisites
The course needs notions of linear algebra, topology and analysis, that is the arguments of the previous courses in Geometry and Analysis.
Course unit content
Geometry of curves and surfaces in the 3D space.
Full programme
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.
Quadrics: definition and examples. Parametrization. Tangent plane. Classification.
Regular surfaces: definition, surfaces preimage of a regular value, surfaces graph of a function and preimage of a regular value. Smooth functions between surfaces. Tangent space and differential of a function. First fundamental form. A characterization of a sphere among compact surfaces. Normal vector field and orientability.
Gauss map's geometry: second fundamental form and curvatures. Geometric meaning of the second fundamental form. Regularity of the curvatures. Hessian of a smooth function. Surfaces of revolution and ruled surfaces. Minimal surfaces.
Intrinsic geometry: isometries. Conformal and area preserving parametrizations. Theorema Egregium. Covariant derivatives and parallel tranport. Geodesics. Classification of surfaces with constant Gauss curvature. Gauss-Bonnet Theorem.
Bibliography
Pdf notes of the course are given.
The lectures follow essentially:
* M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Dover Publications, 2016.
Other sources:
* 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.
* S. Kobayashi, Differential Geometry of Curves and Surfaces, Springer 2019.
Teaching methods
Following the evolution of the sanitary emergency, the lectures could be totally or partially given in streaming, following academic instructions.
Assessment methods and criteria
The final exam, including the first and second part of the Geometry 2 course, consists of a written test and an oral exam. In place of the written exam, students can take two intermediate tests. The evaluation of the intermediate tests and the written test is as follows: students who score between 24 and 30, achieve A.
Students who score between 18 and 23, achieve B. Students who score below 18, achieve C. The written test is considered passed when at least B. Students who achieve at least B in the two intermediate tests have access directly to the oral exam, which can be carried out in any call of the academic year of reference.
The oral exam consists in the proofs of significant theorems and / or in the exposition of topics, definitions, treated in the lectures.
If due to the persistence of the health emergency it was
necessary to integrate with the remote modality the carrying out of the exams will proceed as follows:
remote written tests;
remote oral questions.
Other information
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