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
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Outline

The exercise lectures are divided into two parts:

  1. Review of multivariate calculus and smooth submanifolds in R^n.
  2. 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
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. )