Riemannian geometry 2015

Current affairs Dec 9

The lectures of the first six weeks are now online.

The topics for the final projects are online below.

Contents

The course is an introduction to Riemannian geometry. We will study Riemannian metrics which are positive definite symmetric $(0,2)$-tensors - this means that we endow the tangent space at each point of a smooth manifold with an inner product that varies smoothly. The Riemannian metric is used to define various geometric quantities and objects: the length of a curve, the angle between tangent vectors at a point, parallel transport and geodesics. Parallel transport and geodesics are related with questions such as

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We will also discuss curvature, which "measures how much the Riemannian manifold differs locally from Euclidean space".

Prerequisites

I will assume that the participants are familiar with smooth manifolds and tensors for example from MATS197 Differential geometry.

Lectures

The lectures of the first six weeks are here.

For background material (in Finnish) and recommended books (in English) for differential equations, see here.

Exercises

The problems from the course material for each week are
1 1.2 -1.4, 2.1-2.4 Problem sheet 1 in English
23.2, 3.6-3.8, 4.1-4.2 Problem sheet 2 in English
3 4.3, 4.6, 4.7, 5.1-5.3 Problem sheet 3 in English
4 6.2-6.8 Problem sheet 4 in English
5 7.1-7.7 Problem sheet 5 in English
6 7.9-7.11, 8.1, 8.3, 8.5, 8.7 Problem sheet 6 in English

The topics of the final projects are here.

Grades

Performance is evaluated based on weekly exercises and the final project. Exercises should be returned preferably by email (carefully prepared scanned handwritten solutions or typed).

Literature

Contact information

Jouni Parkkonen
Matematiikan ja tilastotieteen laitos
PL 35
40014 Jyväskylän yliopisto