Geometry 2
- Professors:
- Bonsante Francesco, Pernazza Ludovico
- Year:
- 2015/2016
- Course code:
- 502207
- ECTS:
- 9
- SSD:
- MAT/03
- DM:
- 270/04
- Lessons:
- 84
- Period:
- II semester
- Language:
- Italian
Objectives
Basic knowledge of differential geometry, especially the geometry of curves and surfaces immersed in Euclidean space. Basic knowledge of algebraic topology (fundamental group).
Teaching methods
Lectures and exercise classes.
Examination
Written and oral examination.
Prerequisites
Contents of the courses: Linear algebra, Geometry 1, Algebra 1, Calculus 1 and 2.
Syllabus
Curves
Regular curves. Arc length parameter. Frenet formulae. Curvature and torsion.
Surfaces
Regular surfaces. Diffeomorphisms of surfaces. Tangent plane. First fundamental form. The Gauss map of an orientable surfaces. Second fundamental form. Normal curvature. Gaussian and mean curvature. Isometries and the Theorema Egregium. Geodesics. The Gauss-Bonnet theorem.
Differentiable manifolds
Basic notions and examples. Differentiable maps, tangent and cotangent spaces. The differential. Vector fields and forms.
Fundamental group
Homotopy of paths. Concatenation product and fundamental group. Functorial properties. Deformation retracts. Contractible spaces. Examples and computations.
Regular curves. Arc length parameter. Frenet formulae. Curvature and torsion.
Surfaces
Regular surfaces. Diffeomorphisms of surfaces. Tangent plane. First fundamental form. The Gauss map of an orientable surfaces. Second fundamental form. Normal curvature. Gaussian and mean curvature. Isometries and the Theorema Egregium. Geodesics. The Gauss-Bonnet theorem.
Differentiable manifolds
Basic notions and examples. Differentiable maps, tangent and cotangent spaces. The differential. Vector fields and forms.
Fundamental group
Homotopy of paths. Concatenation product and fundamental group. Functorial properties. Deformation retracts. Contractible spaces. Examples and computations.
Bibliography
M.P. Do Carmo: "Differential Geometry of curves and surfaces", Prentice-Hall.
E. Sernesi: "Geometria 2", Bollati Boringhieri.
C. Kosniowski: "Introduzione alla topologia algebrica", Zanichelli.
C. De Fabritiis, C. Petronio: "Esercizi svolti e complementi di Topologia e Geometria", Bollati Boringhieri.
M. Abate, F. Tovena: "Curve e superfici", Springer.
M. Manetti: "Topologia", Springer.
E. Sernesi: "Geometria 2", Bollati Boringhieri.
C. Kosniowski: "Introduzione alla topologia algebrica", Zanichelli.
C. De Fabritiis, C. Petronio: "Esercizi svolti e complementi di Topologia e Geometria", Bollati Boringhieri.
M. Abate, F. Tovena: "Curve e superfici", Springer.
M. Manetti: "Topologia", Springer.
Modules
- Professor:
- Bonsante Francesco
- Lessons:
- 56
- ECTS:
- 6
- SSD:
- MAT/03
- Professor:
- Pernazza Ludovico
- Lessons:
- 28
- ECTS:
- 3
- SSD:
- MAT/03
Dipartimento di Matematica ''F. Casorati''
Università degli Studi di Pavia -
Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602