Lie Groups and Symmetric Spaces

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Description

The main theme of the lecture is symmetry. Symmetry plays an important role in nature, physics, and hence, in mathematics. Mathematically, symmetry is best described as the action of a group on some set. In the lecture, both, the sets and the groups will have the structure of a smooth manifold, which is the basic object in Differential Geometry. A Lie group is a group that is at the same time a smooth manifold such that group multiplication is smooth. The concept goes back to Sophus Lie and Felix Klein. Manifolds with a Lie group acting transitively on it are quotient spaces that are called homogeneous or symmetric spaces. We will see how geometric properties of these spaces have their correspondence in in algebraic properties of the group.

Objective

The aim of the course is to introduce to the students the notions and techniques of Lie groups, homogeous, and symmetric spaces, that play an important role in Mathematics and Physics. By the end of the course, the student should understand the relations between the algebraic and the geometric notions and be able to perform calculations regarding these spaces.

Content

• Lie groups, their Lie algebras, and the exponential map.
• Homomorphisms, subgroups, and Cartan's Theorem on closed subgroups.
• Homogeneous spaces and symmetric spaces: Riemannian metrics, connections, and curvature.

Thomas Leistner Lecturer for this course

Delivery

2 one-hour lectures per week plus tutorials as required.

Assessment

To be determined.

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)
• Team participation and leadership (7)
• Self-directed study (8)
• Career development (9)

Linkage past

A knowledge of Linear Algebra and basic notions of Differential Geometry is assumed. All other necessary material will developed during the course. To know what a group is and knowledge in Lie Algebras would be helpful but is not mandatory.

Linkage present

No present linkages have been noted.

Linkage future

This course is not recorded as prequisite for other courses.

Restrictions

None.

Recommended text

• F. W. Warner: Foundations of Differentiable Manifolds an Lie Groups
• B. O'Neill: Semi-Riemannian Geometry with Applications to Physics (Chapters 8, 9, and 11)
• S. Helgason: Differential Geometry and Symmetric Spaces (for all)

There are many lecture notes on the internet, for example: J. Berndt, Lie group actions on manifolds.