Geometry (2015/2016)

Course code
Name of lecturer
Giuseppe Mazzuoccolo
Giuseppe Mazzuoccolo
Number of ECTS credits allocated
Academic sector
Language of instruction
II semestre dal Mar 1, 2016 al Jun 10, 2016.

Lesson timetable

II semestre
Day Time Type Place Note
Monday 1:30 PM - 3:30 PM lesson Lecture Hall E  
Wednesday 9:30 AM - 11:30 AM lesson Lecture Hall E  

Learning outcomes

-General Topology.
-Differential geometry of curves.
-Differential geometry of surfaces.


-General Topology.

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. Asymptotes. Contact points of curves. Osculator circle. Main facts about algebraic curves.
Curves in the space:
Tangent line. Normal plane. Inflection points. Osculator plane. Curvatures. Principal frame. Frenet-Serret formula. Torsion.

-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.

Assessment methods and criteria

Written test (2 hours).