Differential geometry and topology (2010/2011)

Course code
Mauro Spera
Academic sector
Language of instruction
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

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.


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