Circles and Spheres

Roughly speaking a Lie Group is both a differentiable (or analytic) manifold and a group, so that the group operations are differentiable (or analytic) maps.

A Symmetric Space is a Riemannian manifold that has an involutive isometry at every point. A Symmetric Space can be written as a quotient manifold of Lie groups, G/H.

This set of notes contains some examples of Lie Groups and Symmetric Spaces.

The style is expository, detailed proofs and caveats are contained in the references.

1. Circles and Spheres as Symmetric Spaces