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

Lectures and exercise classes.

Written and oral examination.

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

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



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.

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.