Differential Geometry – as you need it in Science and Engineering

Aalborg University

Description: Geometry underpins many branches of science and engineering, for example:

• if you wish to describe the motionof 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.

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