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 Geometria 1a-1b, Geometria 2a, Analisi 1a-1b.
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. Regularityof 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 follows essentially:
[1] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Dover Publications, 2016.
Other sources:
[2] M. Abate, F. Tovena, Curve e Superfici, Unitext, Springer, Milano, 2016.
[3] M. Abate, C. de Fabritiis, Geometria analitica con elementi di algebra lineare, McGraw-Hill Education, 2015.
[4] S. Kobayashi, Differential Geometry of Curves and Surfaces, Springer 2019.
Teaching methods
The topics will be formally and rigorously presented through traditional lectures.
Theoretical lectures will alternate with exercises classes, where the students learn how to apply the theory to solve concrete problems.
If the sanitary emergency persists, the lectures could be totally or partially given in streaming, following academic instructions.
Assessment methods and criteria
The exam consists in a written test (two or three exercises to be completed in two and a half hours) and an oral examination which will take place one week later.
Other information
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2030 agenda goals for sustainable development
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