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

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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.

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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!

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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?

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State-Space of Affine Structures on Torus

• Weve 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.

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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

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The End

• Thanks for coming!