Scheda programma d'esame
GEOMETRIA E TOPOLOGIA DIFFERENZIALE
|GEOMETRIA E TOPOLOGIA DIFFERENZIALE||MAT/03||LEZIONI||60|
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KnowledgeThe student who successfully completes the course will be able to demonstrate a solid knowledge of the elementary differential geometry of curves and surfaces, as well as of the rudiments of degree theory in any dimension. He or she will be able to demonstrate familiarity with the various types of curvatures of a surface embedded in the standard Euclidean space, the theorem of Gauss--Bonnet, the hyperbolic plane, the degree of smooth maps between manifolds and important applications such as the fundamental theorem of algebra, Brouwer's fixed point theorem and the hairy ball theorem.
Assessment criteria of knowledgeIn the written test the student must demonstrate his/her knowledge of the course material and to solve simple problems on the course contents. During the oral exam the student must be able to demonstrate his/her knowledge of the course material and be able to discuss the reading material thoughtfully and with propriety of expression.
- Final oral exam
- Final written exam
The examination will consist of two portions: a written test with problems to solve and an oral exam.
Delivery: face to face
- attending lectures
SyllabusCurves in three--dimensional Euclidean space. Arclength parametrization. Frenet frame. Parametrized surfaces and the first fundamental form. The Gauss map and the second fundamental form. The Codazzi and Gauss Equations and Gauss's Theorema Egregium. Covariant differentiation, parallel translation and geodesics. Holonomy and the Gauss--Bonnet theorem. The hyperbolic plane. Manifolds, tangent spaces, smooth maps and their differentials. Regular values. Sard's theorem (without proof). Degree theory. Vector fields and Euler characteristic.
BibliographyTheodore Shifrin “A First Course in Curves and Surfaces”, http://www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf John W. Milnor “Topology from the differentiable viewpoint”, The University Press of Virginia, Charlottesville, 1965.
Ultimo aggiornamento 14/11/2016 17:27