cod. 1007919

Academic year 2020/21
2° year of course - Second semester
Academic discipline
Geometria (MAT/03)
Formazione teorica
Type of training activity
56 hours
of face-to-face activities
6 credits
course unit

Learning objectives

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 unit content

Geometry of curves and surfaces in the 3D space.

Full programme

Conics and quadrics: Definition and examples. Tangent condition and asymptotes. Symmetry's center and axes. 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 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 local theorem.


[1] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Dover Publications, 2016.
[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.

Teaching methods

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 (two or three exercises to be completed in two and a half hours) and an oral examination which will take place in a different date.

Other information

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