Scientific Disciplinary Sector (SSD)
MAT/03 - GEOMETRY
I semestre dal Oct 1, 2019 al Jan 31, 2020.
The course aims to provide students with the basic concepts of the general topology and the basics of differential geometry of curves and surfaces embedded in an Euclidean space. At the end of the course, the student has a general and complete vision of topological properties in a wider context than that of real Euclidean spaces. He/She be able to recognize and compute the main geometrical characteristics of a curve and of a surface (Frenet frames, curvatures, fundamental quadratic forms ...). He/She also be able to produce rigorous arguments and proofs on these topics and he/she can read papers and advanced texts on Topology and Differential Geometry.
The course includes lectures, exercises sessions, and an exam. There will also be 12 hours of tutoring that will focus in particular on the resolution of topology exercises.
Topological space, definition. Examples: trivial topology, discrete topology, discrete topology, cofinite topology. Comparison of topologies. Basis. Neighbourhoods. Closure. Contnuos applications. Homeomorphisms. Limit points and isolated points. Dense set. Topological subspace, induced topology. Product spaces.
Separation axioms. Hausdorff spaces, Normal spaces, Regular spaces.
Countability axioms. Quotient space. Open and closed applications.
Relevant examples: sphere, projective space, Moebius strip...
Compactness. Heine-Borel Theorem. Tychonoff Theorem. Bolzano-Weierstrass Theorem.
Connectivity, local connectivity. Path connectivity. Examples and counterexamples. Simply connected, homotopy and fundamental group. Jordan curve Theorem.
-Differential geometry of curves.
Curves in the plane:
Examples. Regular points and singular points. Embedding and immersion. Vector fields along a curve. Tangent vector and line. Length of an arc. Parametrization by arc-length. Inflection points. Curvature and radius of curvature. Center of curvature. Frenet-Serret formula.
Curves in the space:
Tangent line. Normal plane. Inflection points. Osculator plane. Curvatures. Principal frame. Frenet-Serret formula. Torsion. Fundamental Theorem.
-Differential geometry of surfaces.
Definitions. Differentiable atlas. Oriented atlas, Tangent plane, Normal versor.
First fundamental quadratic form: metric and area. Tangential curvature and normal curvature of a curve on a surface. Curvatures, normal sections, Meusnier Theorem. Principal curvatures, Gaussian curvature and mean curvature: Theorem Egregium. Geodetics.
||Curve e Superfici
|| Introduzione alla topologia algebrica
To pass the exam, students must show that:
- they know and understand the fundamental concepts of general topology
- they know and understand the fundamental concepts of local theory of curves and surfaces
- they have analysis and abstraction abilities
- they can apply this knowledge in order to solve problems and exercises and they can rigorously support their arguments.
Written test (2 hours).
The exam consists of four exercises (2 on topology, 1 on curve theory and 1 on surfaces theory) and two questions (1 on general definition / concepts and 1 with a proof of a theorem presented during the lectures).
Oral Test (Optional)
It is a discussion with the lecturer on definitions and proofs discussed during the lessons.