Lie Groups, Symmetric Spaces, Curves, Surfaces

Lie Groups and Symmetric Spaces

No reckoning allowed

save the marvelous arithmetics

of distance

 Audre Lorde, https://en.wikipedia.org/wiki/Audre_Lorde

 

“There is an exception to every rule, including this one.”

                                                            Basque Proverb

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.  Please click on the links for the pdf. (S. Peterson,  batugeo@gmail.com)

1.    Circles and Spheres as Symmetric Spaces

4. Orientation on Surfaces

Animations and writeup, click the link above.

6. Cassini Ovals


7, Geodesics on the Torus


8. Parabola into cardioid



14. Saddle