Differential geometry and topology (2009/2010)

Course code
4S02812
Name of lecturers
Mauro Spera, Nicola Sansonetto
Coordinator
Mauro Spera
Number of ECTS credits allocated
12
Academic sector
MAT/03 - GEOMETRY
Language of instruction
Italian
Period
1st Semester dal Oct 1, 2009 al Jan 31, 2010.

Lesson timetable

1st Semester
Day Time Type Place Note
Monday 2:30 PM - 4:30 PM lesson Lecture Hall M  
Tuesday 2:30 PM - 4:30 PM lesson Lecture Hall M  
Wednesday 2:30 PM - 4:30 PM lesson Lecture Hall M  
Thursday 2:30 PM - 4:30 PM lesson Lecture Hall M  

Learning outcomes

Learning objectives

The course delves further into general topology and introduces the basic notions of algebraic
and differential topology, focussing on the concept of differentiable manifold. Furthermore, the
elements of Riemannian geometry will be introduced as well.
The course, suitable to both curricula (didactic and applied) will be quite concrete and based
on examples also coming from other areas of mathematics.

Syllabus

Course Programme

General topology (continued). Separation. Quotients.
Fundamental group. Covering spaces.
Differentiable manifolds.
De Rham's theory.
Riemannian manifolds.
Levi-Civita connection.
Curvature tensors (Riemann, sectional, Ricci, scalar).
Geodesics and their variational aspects.
Exponential map.
Lie groups. Symmetric spaces.
Riemann surfaces and algebraic curves.
Vector bundles, Euler's class and number, Euler-Poincare' characteristic.
The Poincare'-Hopf theorem.

Assessment methods and criteria

Written test, followed by an oral exam.

Teaching aids

Documents