Course Unit Page

Academic Year 2021/2022

Learning outcomes

At the end of the course, the student acquires advanced knowledge on smooth manifolds and differential calculus with particular regard to de Rham cohomology and Morse theory. It is able to apply the acquired notions for solving problems and building demonstrations.

Course contents

Differentiable manifolds.
Differentiable functions.
Introduction to transversality.
Morse and Sard theorems.
Tangent space to a manifold.
Vector fields.
Distributions and Frobenius theorem.
Vector bundles.
Differential forms.
Exterior derivative.
Introduction to Riemannian geometry.
orientation and Stokes theorems.
De Rham cohomology.


L. Tu, Introduction to manifolds.

F. Warner, Foundations of differentiable manifolds and Lie groups.

J. M. Lee, Introduction to smooth Manifolds, (GTM Springer).

Teaching methods

Frontal Lessons

Assessment methods

The exam related to this component of the integrated course (00474 - DIFFERENTIAL GEOMETRY - 6 credits) consists of an oral test.

The exam mark attributed to the student is given by the average of the marks awarded for the two modules of the course, with rounding to the upper unit. The mark 29 does not preclude 30L as final mark.

Office hours

See the website of Sergio Venturini