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)
Animations and writeup, click the link above.
5. Cornu Spiral
6. Cassini Ovals
7, Geodesics on the Torus
8. Parabola into cardioid
12. Poincare Disc
14. Saddle
17. Hypocycloids