Geometry on Groups

Matrix groups as examples of Lie groups. Lie algebras and their description via invariant differential forms. Local coordinates on matrix groups. Invariant Riemannian metrics. Connections and curvature via Lie algebras. Rough classification of Lie algebras: semi-simple, nilpotent and solvable algebras. Flat groups; constraints on Ricci curvature. Use of elementary techniques from representation theory. Hermitian structures, SKT structures, particular geometries in dimensions up to six. Classification theorems, particularly on nilpotent and solvable groups.

Objectives of the course:
Geometric structures such as metrics or complex structures are important objects, but it is often hard to find examples that solve particular equations. Adding symmetry to problems can provide a dramatic simplification, and considering problems where the underlying space is a group reduces many questions to those of (multi-) linear algebra. The aim of this course is to introduce geometry on matrix groups from the linear algebra perspective. This will then be used to provide a number of concrete constructions of Einstein metrics and to give classifications for other geometric structures, particularly in low dimensions.

Learning outcomes and competences:
At the end of the course the students should be able to:

• reproduce key results for geometric structures on matrix groups and give rigorous and detailed proofs of them,
• apply the basic techniques and concepts of the course to concrete examples and use them to show how classification results may be derived,
• combine concepts from linear algebra and geometry to work with invariant geometric structures defined via tensors and differential forms,
• relate invariant concepts on matrix groups to geometric objects on submanifolds of Euclidean space, and
• given an oral presentation of some topic from the course.

Course homepage:
http://bb.au.dk/

Blackboard for students, - get help here:
http://studerende.au.dk/en/selfservice/blackboard/