Welcome to Differential Geometry – as you need it in Science and Engineering**Description:** Geometry underpins many branches of science and engineering, for example:

- if you wish to describe the motion of an airplane or of a robot arm

- if you look closer at the design of the GPS navigation system in a car

- if you analyse the control of a telecommunication satellite

- if you want to exploit the options and understand the constraints you meet in customary programmes in computer graphics, computer vision or computer games

- if you study the configurations of a mechanical or molecular system, or problems in relativity theory or in elementary particle physics.

It is quite common that the underlying configuration space of a system is not some Euclidean space Rn, but a manifold that only locally can be described by a fixed set of coordinates. The number of coordinates required corresponds to the degrees of freedom of the system. Familiar examples of manifolds are spheres

- in arbitrary dimensions - and tori - in 2 dimensions visualized as a donut. Other typical examples occur as the space of orthogonal matrices – the configurations of a mechanical system with a fixed point; in dimension 3, these are related to the so-called Euler angles. Similarly, projective space is the space of directions of a 1-dimensional rod fixed at a point.

How can one describe such a manifold, how do coordinates change? What corresponds to velocity vectors?

What is the counterpart of a differential equation on such a gadget, under which conditions can it be solved? What types of symmetries are there on (some of) these manifolds, and how to exploit them? …

In mathematical terms, subjects covered by the course will include

- Manifolds: concept, definition and examples

- Tangent vector and tangent bundle

- Differentiable maps between manifolds and their differentials

- Inverse and implicit function theorem as a tool to uncover and to analyse manifolds

- Vector fields, flows, Lie brackets

Other possible topics to be chosen from and depending on the audience:

- Differential forms and Stokes’ Theorem

- Distributions and the Frobenius Theorem

- Riemannian metrics

- Curvature concepts

The lectures will incorporate examples of the use of notions and results from differential geometry in engineering and science; the choice of examples will depend on the audience.

Prerequisites:

Solid background in mathematical techniques as obtained through engineering studies at Aalborg University, is expected.

Key literature:

L. W. Tu, An Introduction to manifolds, Second Edition, 2011, Springer,

New York.**Organizer:** Lisbeth Fajstrup, Associate Professor, e-mail: fajstrup@math.aau.dk and Martin Raussen, Associate Professor, email: raussen@math.aau.dk**Lecturers:** Associate Professor, Lisbeth Fajstrup, Associate Professor

Martin Raussen and Associate Professor Rafael Wisniewski**ECTS:** 4**Time:** 2, 4, 9, 10, 11 and 16 December, 2014**Place:** Aalborg University, Niels Jernes Vej 12A/6-104**Zip code: **9220**City: **Aalborg**Number of seats: **40**Deadline: **1 December, 2014

- Teacher: Lisbeth Fajstrup
- Teacher: Martin Raussen
- Teacher: Rafal Wisniewski