Mikko Salo

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Differential geometry on manifolds (MATS1970), period 2, 2025

This course is a continuation of Introduction to manifolds from period 1. We will cover the cotangent bundle of a manifold, differential 1-forms, tensor bundles and tensor fields, and differential k-forms. The course will conclude with a discussion of integration on manifolds. This sequence of differential geometry courses will continue with Geometry of geodesics in period 3.

Instructors

The lectures are given by Mikko Salo (office MaD359), and exercise sessions by William Trad (office MaD345). You are very welcome to contact the instructors at their offices or by email.

Schedule

The lectures are Mondays at 12.15-14.00 and Tuesdays at 10.15-12.00 in room MaD380 (27 Oct-2 Dec).

Exercise sessions are Mondays at 14.15-16.00 in MaD355 (3 Nov-8 Dec). Please return your answers for each exercise sheet to Moodle by Friday midnight.

Material

Luentomuistiinpanot (by Jouni Parkkonen, in Finnish), Lecture notes (AI-assisted translation, to be updated)

In this course, we will cover part II (chapters 6-10) of the lecture notes. Further material will be provided on the Moodle page.

As additional reading, one can use the the relevant parts of the following textbook (which the lecture notes are based on):

• J. M. Lee. Introduction to smooth manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, second edition, 2013.

Prerequisites

Introduction to manifolds.

Completion

The course can be taken for credit (4 cr) by attending the lectures and exercises, by returning written answers to exercises, and by taking the course exam on 10 Dec (retake on 26 Jan). The course exam will count for 70 % of the grade and the exercises for 30 %. Half of the maximum points are required to pass, and grading will be on the scale 1-5.

Please register via Sisu.