Exercise class (15h) for the master course "Istituzioni di Geometria Superiore".
| Where: |
Aula 29,
(h: 15:30 - 17:30 ) Via Garzetta 48 |
| When: |
AY 2023-24,
(winter semester) |
| Contacts | Dr. Antonio Miti |
Outline
The exercise lectures are divided into two parts:
- Review of multivariate calculus and smooth submanifolds in R^n.
- Exercises of Cartan calculus taken from the past exam sheets.
The first part is taken from Prof. Spera's lectures notes (diffgeotopo-I-II,
III,
IV).
Alternatively, the same material has been covered by the lectures notes of the previous tutors: Andrea Galasso and Micol Campagnaro.
All past exam sheets can be found at the following link.
Bibliography
- M. Spera, Differential geometry and topology, lecture notes, available online on the course webpage.
- M. Campagnaro, Esercitazioni di istituzioni di geometria superiore, lecture notes, available online on Unicatt-Blackboard, folder "Esercitazioni-Dispense (Campagnaro)".
- A. Galasso, Review of calculus and introduction to smooth manifolds, lecture notes, available online on the course webpage.
- J.M. Lee, Introduction to smooth manifolds, Springer-Verlag, Berlin, Heidelberg, New York, 2003.
- M. Abate and F. Tovena, Geometria differenziale, Springer-Verlag Italia, Milano, 2011.
Syllabus
| Thu 19/10/22 | Review on multivariable calculus.
Smooth functions, Differential and interpretations. [Lecture notes] ( See also: Spera, diffgeotopo-I and Galasso, section 1. ) |
| Wed 25/10/22 | Implicit functions and embeddings. (in Aula 25)
Inverse Function Theorem. Constant rank theorem. Dini's theory of implicit functions. Immersions, submersions and embeddings in R^n with examples. [Lecture notes] ( See also: Spera, diffgeotopo-II and Galasso, sections 2,3,4. ) |
| Thu 26/10/22 | Submanifolds of the Euclidean space and Tangent spaces
Parametrized surfaces in R^3. Submanifolds of dimension n in R^(n+m). Tangent space of a submanifold. [Lecture notes] ( See also: Spera, diffgeotopo-III and Galasso, section 5. ) |
| Thu 09/11/22 | Examples of Submanifolds and Lie groups
Examples: 1-dimensional submanifolds in R^2. graphs, spheres, torii, hyperboloids. Smooth structure and tangent spaces of SO(n). [Lecture notes] [Addendum] ( See also: Spera, diffgeotopo-IV. ) |
| Thu 16/11/22 | Lie derivative and Cartan calculus
Local representation of tensor fields, Lie derivative, calculus on differential forms. [Lecture notes] [Cheatsheet] [Cheatsheet on Github] ( See also: Spera, diffgeotopo-XXIII. ) |
| Thu 23/11/22 | Lie brackets and Integrable distributions
Explicit computation of the Lie bracket, integrability of smooth distributions. [Lecture notes] [Addendum] ( See also: Spera, diffgeotopo-XXV. ) |
| Thu 30/11/22 | Symplectic Manifolds
Closed and exact differential forms, symplectic forms, Hamiltonian fields, pullback of differential forms. [Lecture notes] ( See also: Spera, diffgeotopo-XXVII. ) |
| Thu 7/12/22 | Riemannian Manifolds
Covariant derivative, Riemannian geometry, Killing fields, curvature. [Lecture notes] ( See also: Spera, diffgeotopo-XXXI. ) |