Course Syllabus for

# Riemannian Geometry Riemanngeometri

## FMA160F, 7.5 credits

Valid from: Autumn 2018
Decided by: Professor Thomas Johansson
Date of establishment: 2018-10-08

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: MATM23
Teaching language: English

## Aim

The aim of the course is to acquaint the postgraduate student with the basics of Riemannian geometry, in particular smooth manifolds of arbitrary finite dimensions, tangent bundles and Lie derivatives. The subject is an important field of research in mathematics, but methods from the subject are also important for mechanics -- where typically the phase space for a mechanical system is described by the tangent bundle of a nontrivial manifold -- and in the general theory of relativity.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to account for the definition a differentiable manifold and be able to describe the most common families of Lie groups.
• be able to account for the definition of the tangent space of a differentiable manifold, and give examples.
• be able to explain the concepts immersion, embedding and submersion, and state different versions of the inverse mapping theorem.
• be able to account for the definition of the tangent bundle of a differentiable manifold, with examples, and explain the relation between Lie algebras and Lie groups.
• be able to account for the definitions of a Riemann metric and of a Riemannian manifold, and give some examples.
• be able to account for the definition of the Levi-Civita connection on a Riemannian manifold, and explain why it is useful.
• be able to briefly account for the theory of geodesics on Riemannian manifolds, and for polar coordinates.
• be able to state some properties of the Riemann curvature tensor.

## Course Contents

Differentiable Manifolds. The Tangent Space. The Tangent Bundle. Riemannian Manifolds. The Levi-Civita Connection. Geodesics. The Riemann Curvature Tensor. Curvature and Local Geometry.

## Course Literature

Gudmundsson, S.: An Introduction to Riemannian Geometry. Centre for Mathematical Sciences, Lund University, 2017.
The participants should also consult some of the other books that are recommended on the course web page.

## Instruction Details

Type of instruction: Lectures

## Examination Details

Examination format: Oral exam