Differential geometry and topology (2011/2012)

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

Educational objectives

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.


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

Exam methods

Written test, followed by an oral exam.

Teaching aids
Title Format (Language, Size, Publication date)
add-Erlangen  pdfpdf (it, 51 KB, 05/12/11)
addXXXII-11-12  pdfpdf (it, 55 KB, 29/11/11)
hyp-addXXXII  pdfpdf (it, 117 KB, 29/11/11)
programma ufficiale topogeo -2011/12  pdfpdf (it, 43 KB, 11/01/12)
topogeo-add-I 11-12  pdfpdf (it, 35 KB, 05/10/11)
topogeo-add-II 11-12  pdfpdf (it, 72 KB, 18/10/11)
topogeo-add-III 11-12  pdfpdf (it, 57 KB, 20/10/11)
topogeo-add-XVII 11-12  pdfpdf (it, 75 KB, 03/11/11)
topogeo-add-XXI 11-12  pdfpdf (it, 87 KB, 08/11/11)
topogeo-add-XXIII 11-12  pdfpdf (it, 38 KB, 14/11/11)
topogeo-add-XXV 11-12  pdfpdf (it, 50 KB, 21/11/11)
topogeo-add-XXVII 11-12  pdfpdf (it, 85 KB, 23/11/11)
topogeo-scritto 17-9-12  pdfpdf (it, 146 KB, 18/09/12)
topogeo-scritto 18-6-12  pdfpdf (it, 155 KB, 18/06/12)
topogeo-scritto 20-2-12  pdfpdf (it, 172 KB, 20/02/12)
topogeo-scritto 6-2-12  pdfpdf (it, 282 KB, 06/02/12)
topogeo-scritto 9-7-12  pdfpdf (it, 126 KB, 09/07/12)