# Differential geometry and topology (2010/2011)

Course code
4S02812
Credits
12
Coordinator
Mauro Spera
MAT/03 - GEOMETRY
Language of instruction
Italian
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Teoria 10 I semestre Mauro Spera
Esercitazioni 2 I semestre Nicola Sansonetto

### Lesson timetable

I semestre
Activity Day Time Type Place Note
Teoria Monday 2:30 PM - 4:30 PM lesson Lecture Hall M
Teoria Tuesday 1:30 PM - 3:30 PM lesson Lecture Hall M
Teoria Wednesday 2:30 PM - 4:30 PM lesson Lecture Hall M
Teoria 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.

 Title Format (Language, Size, Publication date) note sansonetto pdf (it, 354 KB, 04/02/11) programma ufficiale topogeo 2010/11 pdf (it, 45 KB, 17/01/11) topogeo-16-2-11 pdf (it, 206 KB, 16/02/11) topogeoI pdf (it, 432 KB, 12/09/10) topogeo-scritto 1-7-11 pdf (it, 200 KB, 07/07/11) topogeo-scritto 19-9-11 pdf (it, 209 KB, 19/09/11) topogeoXLI pdf (it, 355 KB, 12/01/11) topogeoXLII pdf (it, 445 KB, 19/01/11) topogeoXLIII pdf (it, 371 KB, 19/01/11) topogeoXLIV pdf (it, 304 KB, 19/01/11) topogeoXXIX-add pdf (it, 30 KB, 24/11/10) topogeoXXVI-add pdf (it, 91 KB, 22/11/10) topogeoXXXII-add pdf (it, 196 KB, 30/11/10)