# Differential geometry and topology (2012/2013)

Course code
4S001100
Credits
12
Coordinator
Gaetano Zampieri
MAT/03 - GEOMETRY
Language of instruction
Italian
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Modulo 1 6 I semestre Mauro Spera
Modulo 2 6 II semestre Gaetano Zampieri

### Learning outcomes

Module: 1
Learning objectives:

The course provides an introduction to differentiable manifolds
and Riemannian geometry; it will possess a quite concrete character and will be based
on examples emerging from other areas of mathematics as well.

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### Syllabus

Module: 1
Syllabus:
Multilinear algebra.
Differentiable manifolds.
Lie groups.
Tensor calculus. Lie group actions on manifolds and their orbit spaces.
Riemannian manifolds.
Levi-Civita connection.
Curvature tensors (Riemann, sectional, Ricci, scalar).
Geodesics and their variational aspects.
Exponential map.
Riemannian geometry of Lie groups.

Module: 2

First Part:

0) Integration on manifolds and Stokes Theorem

1) Riemannian Geometry: connections, parallel transport, curvature
tensors, Levi-Civita connection, geodesics and applications (Chap. 1 and 2
of Chavel's Book "Riemannian Geometry, a modern Introduction" (2nd
Edition, Cambridge University Ppress, 2006).

2) Morse theory: Morse lemma and applications ( J. Milnor "Morse Theory" ,
Annals of Mathematics Studies, Princeton University Press).

3) De Rham cohomology and applications (Chap.1 of Bott, Tu "Differential
Forms in Algebraic Topology" , Graduate Texts in Mathematics, 82).

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### Assessment methods and criteria

Exam methods (1st module): written test, immediately followed by an oral exam. The final grade will be assigned after completion of the second module.

 Title Format (Language, Size, Publication date) diffgeotopo-19-2-13 pdf (it, 150 KB, 19/02/13) diffgeotopo-I pdf (it, 281 KB, 31/10/12) diffgeotopo-II pdf (it, 3914 KB, 03/11/12) diffgeotopo-III pdf (it, 2598 KB, 03/11/12) diffgeotopo-IV pdf (it, 221 KB, 06/11/12) diffgeotopo-IX pdf (it, 390 KB, 16/11/12) diffgeotopo-V pdf (it, 225 KB, 09/11/12) diffgeotopo-VI pdf (it, 257 KB, 09/11/12) diffgeotopo-VII pdf (it, 347 KB, 12/11/12) diffgeotopo-VIII pdf (it, 188 KB, 16/11/12) diffgeotopo-X-1-bad pdf (it, 1350 KB, 19/11/12) diffgeotopo-X-2-bad pdf (it, 649 KB, 19/11/12) diffgeotopo-XI pdf (en, 294 KB, 23/11/12) diffgeotopo-XII pdf (en, 550 KB, 26/11/12) diffgeotopo-XIII pdf (en, 285 KB, 17/12/12) diffgeotopo-XIII-corrections pdf (it, 93 KB, 11/04/13) diffgeotopo-XII-p.15 pdf (it, 43 KB, 30/11/12) diffgeotopo-XIV pdf (en, 559 KB, 17/12/12) diffgeotopo-XIX pdf (en, 304 KB, 14/12/12) diffgeotopo-XV pdf (en, 491 KB, 30/11/12) diffgeotopo-XVI pdf (en, 200 KB, 03/12/12) diffgeotopo-XVII pdf (en, 493 KB, 07/12/12) diffgeotopo-XVIII-1 pdf (en, 323 KB, 10/12/12) diffgeotopo-XVIII-2 pdf (en, 181 KB, 10/12/12) diffgeotopo-XX pdf (en, 545 KB, 21/12/12) diffgeotopo-XX-addendum pdf (it, 30 KB, 28/05/13) diffgeotopo-XX-addendum2 pdf (it, 30 KB, 28/05/13) official-programme-1st module 12/13 pdf (en, 37 KB, 20/12/12)