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

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

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

Geometric Structure To construct a geometric structure: To construct a geometric structure: Start with a surface S Start with a surface S Cut it Cut it Embed into X Embed into X Surface inherits the geometric structure of X Surface inherits the geometric structure of X For X = R 2, G = Isom(R 2 ), S = square torus - this construction defines a Euclidean structure on the torus. For X = R 2, G = Isom(R 2 ), S = square torus - this construction defines a Euclidean structure on the torus.

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

Construct an Affine Structure on the Torus Affine group, Aff(R 2 ) consists of maps of R 2 to itself which carry lines to lines. Affine group, Aff(R 2 ) consists of maps of R 2 to itself which carry lines to lines. Consider an arbitrary quadrilateral with identifications which yield a torus. Consider an arbitrary quadrilateral with identifications which yield a torus. Embed into R 2 Embed into R 2 Use Affine maps, A & B as transitions. Use Affine maps, A & B as transitions. A & B represent elements of the fundamental group of the torus. A & B represent elements of the fundamental group of the torus. Is the Euclidean metric preserved? Is the Euclidean metric preserved?

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

Other Surfaces & Structures Surfaces of genus 2 have Hyperbolic Structure. Geometry is (H 2,PSL 2 (R)). Surfaces of genus 2 have Hyperbolic Structure. Geometry is (H 2,PSL 2 (R)). Any Hyperbolic structure gives Projective structure, modeled on (CP 1, PSL 2 (C)). Any Hyperbolic structure gives Projective structure, modeled on (CP 1, PSL 2 (C)). Here we see different geometric structures with the same representation of the fundamental group. Interesting… Here we see different geometric structures with the same representation of the fundamental group. Interesting…

The End Thanks for coming! Thanks for coming!