Enrollment year
2019/2020
Academic discipline
MAT/03 (GEOMETRY)
Department
DEPARTMENT OF MATHEMATICS "FELICE CASORATI"
Curriculum
PERCORSO COMUNE
Period
2nd semester (01/03/2021 - 11/06/2021)
Lesson hours
84 lesson hours
Activity type
WRITTEN AND ORAL TEST
Prerequisites
Contents of the courses: Linear algebra, Geometry 1, Algebra 1, Calculus 1 and 2.
Learning outcomes
Basic knowledge of differential geometry of curves and surfaces immersed in Euclidean space. Basic knowledge of algebraic topology (fundamental group).
Course contents
Differential geometry of immersed curves and surfaces. 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.
Fundamental group
Homotopy of paths. Concatenation product and fundamental group. Functorial properties. Deformation retracts. Contractible spaces. Examples and computations.
Teaching methods
Lectures and exercise classes.
Reccomended or required readings
M.P. Do Carmo: "Differential Geometry of curves and surfaces", Prentice-Hall.
E. Sernesi: "Geometria 2", Bollati Boringhieri.
M. Manetti: "Topologia", Springer.
Assessment methods
Written and oral examination.
The written exam consists of exercises on the topics of course. The oral exam verifies the knowledge and understanding of the definitions and of the theorems.
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