2015/2016

2016/2017

DM270

MAT/03 (GEOMETRY)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

2nd semester (01/03/2017 - 09/06/2017)

9

84 lesson hours

ITALIAN

ORAL TEST

BONSANTE FRANCESCO (titolare) - 6 ECTS
FREDIANI PAOLA - 3 ECTS

Contents of the courses: Linear algebra, Geometry 1, Algebra 1, Calculus 1 and 2.

Basic knowledge of differential geometry, especially the geometry of curves and surfaces immersed in Euclidean space. Basic knowledge of algebraic topology (fundamental group).

Differential geometry of immersed curves and surfaces. Differentiable manifolds. Fundamental group.

Extended summary

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.

Lectures and exercise classes.

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.


M. Abate, F. Tovena: "Curve e superfici", Springer.


M. Manetti: "Topologia", Springer.

Written and oral examination.

Written and oral examination.