Points in the State Space of a Topological Surface

1. Points in the State Space of a Topological Surface Josh Thompson February 28, 2005

2. Defining Geometry • 1872 – Felix Klein • “Geometry is the study of the properties of a space which are invariant under a group of transformations.”

3. Model Geometry • Model Geometry = a pair (X,G) • X = simply connected manifold (topological space locally ‘similar’ to Rn with no ‘holes’) • G = Group of transformations of X, acting transitively on X

4. Geometric Structure • To construct a geometric structure: • Start with a surface S • Cut it • Embed into X • Surface inherits the geometric structure of X • For X = R2, G = Isom(R2), S = square torus - this construction defines a Euclidean structure on the torus.

5. Can we construct other Euclidean structures on the torus? • Define the state of the surface to be a particular geometric structure. • The state space of Euclidean structures on the torus can be identified with the upper half plane. • Yes, we can!

6. Construct an Affine Structure on the Torus • Affine group, Aff(R2) consists of maps of R2 to itself which carry lines to lines. • Consider an arbitrary quadrilateral with identifications which yield a torus. • Embed into R2 • Use Affine maps, A & B as transitions. • A & B represent elements of the fundamental group of the torus. • Is the Euclidean metric preserved?

7. State-Space of Affine Structures on Torus • We’ve constructed a “valid” geometry; however, it has no notion of distance. Cool. • State-space in the Affine case is much larger than the Euclidean case. • There are many more Affine structures on the torus than Euclidean structures.

8. Other Surfaces & Structures • Surfaces of genus 2 have Hyperbolic Structure. Geometry is (H2,PSL2(R)). • Any Hyperbolic structure gives Projective structure, modeled on (CP1, PSL2(C)). • Here we see different geometric structures with the same representation of the fundamental group. Interesting…

9. The End • Thanks for coming!