Lie Groups, Symmetric Spaces, Curves, Surfaces

Fermat's Last Theorem, n=3. A Self-contained Proof.

Cubic Surface x^3+y^3=z^3+u^3 

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.

5. Cornu Spiral

https://en.wikipedia.org/wiki/Euler_spiral

6. Cassini Ovals

7. Enneper Surface

7, Geodesics on the Torus

8. Parabola into cardioid

9. Ellipses and Ellipsoids

10. Minimal Surfaces - Transform Helicoid into Catenoid

11. Stereographic Projection

12. Poincare Disc

13. Involutes and Evolutes

14. Saddle

15, Whitney Umbrella with Unit Normal Map

16. Trochoid and Cycloid

17. Hypocycloids

18. Semi-circle with diameter homotopic to a triangle

11. Construct a Parabola using Compass and Straightedge

12. Construct an Ellipse

13. Another construction of an ellipse

14. Inversions in the Circle and Steiner's Porism