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DIFFERENTIAL GEOMETRY - 8 CFU

Teacher

Davide Barilari

Scheduled Period

I Year - 1 Semester | 04/10/2021 - 15/01/2022

Hours: 64 (32 esercitazione, 32 lezione)

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Prerequisites

The course requires notions of Linear Algebra (vector spaces, linear maps, matrices, bilinear forms, and more in general multilinear forms) and Analysis (differential and integral calculus for real functions of one or more variables). Some knowledge of general topology is also required (open and closed sets, connectedness, compactness and main properties).

Target skills and knowledge

At the end of the course it is required that the student is familiar with the notion of differentiable manifold and can apply, both in theory and exercices, differential and integral calculus on manifolds. It is required that the student knows the proof of the main results of the course and can apply such notions to solve some concrete geometric problems.

Examination methods

The exam is based on a written test and an oral test. During written test the student is asked to solve some exercises where he must be able to apply theoretical notions studied during the course. The student who is admitted to the oral test will be asked to answer to some questions on the main notions and results about the course. The final mark is based on the results of both the written test and the oral test.

Assessment criteria

The student will be evaluated in terms of
1. Global understanding of the notions introduced in the course;
2. Clarity in explaining the mathematical notions, together with the appropriate mathematical language
3. Ability to apply theoretical notions to solve some concrete problems

Course contents

1. Differentiable manifolds. Smooth maps between manifolds.
Tangent space to a manifold, tangent vectors and derivations.
Submanifolds: immersions, submersions and embeddings.
2. Tangent bundle. Vector fields and flows. Lie brackets.
Distributions, integrable and non integrable distributions, Frobenius theorem.
3. Cotangent bundle, vector bundles. Tensors.
Differential forms on a manifold, external algebra.
Orientable manifolds and manifolds with boundary.
Integration of differential forms. Stokes theorem.
4. Metrics on a manifold: Riemannian manifolds. Geodesics.
Linear connections compatible with a metric. Levi-Civita connection.
Parallel transport. Riemann curvature tensor.

Planned learning activities and teaching methods

Teaching activity is based on lesson in class.

Additional notes about suggested reading

All supplementary material, such as lecture notes and suggested textbooks, will be available in the moodle page of the course at elearning.unipd.it/math/

Textbooks (and optional supplementary readings)

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