Project IV (MATH 4072) 2019-2020

Geometric Structures on Manifolds

Anna Felikson

Description:

Take a square on Euclidean plane and identify the opposite sides. You will obtain a torus with additional geometric structure: it will look the same as Euclidean plane locally near every point. One can ask which surfaces admit geometric structures of this type. It turns out, that depending on the type of surface one can always turn the surface into spherical, Euclidean or hyperbolic one (in most cases, it will be a hyperbolic structure).

One can ask a similar question about manifolds of higher dimensions: when a manifolds admits a geometric structure? What kind of geometries should one use as building blocks?

In this project we will look at the interaction of geometry and topology of surfaces and manifolds of higher dimension. Possible topics may include the following:
  • Geometries on surfaces
  • Consruction of geometric structures on surfaces (by triangulations, by pants decompositions)
  • Building blocks for 3-manifolds: eight 3-dimensional geometries
  • Decomposition of 3-manifolds into geometric pieces
  • Dehn fillings
  • Affine and projective structures on manifolds
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Prerequisites: Geometry III, Algebra II
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Resources:

One can look at the following books or lecture notes: One can also look at the following webpages:

email: anna dot felikson at durham dot ac dot uk

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