IstGeoSup 22-23

Exercise class (15h) for the master course "Istituzioni di Geometria Superiore".

Where:	Aula 23, Via Garzetta 48
When:	AY 2022-23, (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 13/10/22	Review on multivariable calculus. Differential and interpretations. Inverse Function Theorem. [Lecture notes] ( See also: Spera, diffgeotopo-I and Galasso, section 1. )
Thu 20/10/22	Implicit functions and embeddings. 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 27/10/22	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. 1-dimensional submanifolds in R^2. [Lecture notes] ( See also: Spera, diffgeotopo-III and Galasso, section 5. )
Thu 03/11/22	Examples of Submanifolds and Lie groups Examples: graphs, spheres, torii, hyperboloids. Smooth structure and tangent spaces of SO(n). [Lecture notes] ( See also: Spera, diffgeotopo-IV. )
Thu 10/11/22	Lie derivative and Cartan calculus Local representation of tensor fields, Lie derivative, calculus on differential forms. [Lecture notes] [Cheat sheet] ( See also: Spera, diffgeotopo-XXIII. )
Thu 17/11/22	Lie brackets and Integrable distributions Explicit computation of the Lie bracket, integrability of smooth distributions. [Lecture notes] ( See also: Spera, diffgeotopo-XXV. )
Thu 24/11/22	Symplectic Manifolds Closed and exact differential forms, symplectic forms, Hamiltonian fields, pullback of differential forms. [Lecture notes] ( See also: Spera, diffgeotopo-XXVII. )
Thu 01/12/22	Riemannian Manifolds Covariant derivative, Riemannian geometry, Killing fields, curvature. [Lecture notes] ( See also: Spera, diffgeotopo-XXXI. )