# Geometry III/IV

## Epiphany 2013

 Time and place: Fr 13:00, 15:00 CG60 Instructor: Anna Felikson e-mail: anna dot felikson at durham dot ac dot uk Office: CM235; Phone: 334-4158 Office hours: Mon 12:00-13:00

 Textbook:

• V. V. Prasolov, Non-Euclidean Geometry
• (this book was distributed in class, thanks to the kind permission of the author and the publisher).

• E. B. Vinberg (Ed.), Geometry II.
• (Chapter 3 of Part I: Geometry of Spaces of Constant Curvature ), Encyclopaedia of Mathematical Sciences, Vol. 29, Springer-Verlag.
• H. S. M. Coxeter, Introduction to Geometry, Wiley, published 1963.
• S. Katok, Fuchsian Groups, University of Chicago Press, 1992.
• E. Rees, Notes on Geometry, Universitext, Springer, 2004.
• Preliminary course content (subject to change): spherical geometry, Möbius transformations, hyperbolic geometry.

Schedule:

• Week 11: Spherical geometry: spherical lines, triangle inequality, polar correspondence; sperical triangles (congruence, area, law of sines, law of cosines).
• Week 12: Spherical geometry: law of sines, law of cosines; isometries of the sphere. Möbius transformations: definition, triple transitivity, fixed points and conjugacy classes.
• Week 13: Inversions: take circles and lines to circles and lines, preserve anlges. Möbius transformation is a composition of even number of inversions. Cross-ratio: cross-ratios are preserved by Möbius transformations, cross-ratio is real if and only four points lie on one line or circle. Inversion in space.
• Week 14: Stereographic projection. Introduction to hyperbolic geometry: Poincare disk model. Problem session at the second hour (overview of easy exercises on spherical geometry and Möbius transformations).
• Week 15: Conformal models of hyperbolic plane: Poincare disk and half-plane models. Hyperbolic lines, circles. Isometry group.
• Week 16: Elementary hyperbolic geometry: parallel lines, angle of parallelism, hyperbolic triangle (sum of angles, congruence of triangles, hyperbolic trigonometry, law of sines, law of cosines, area of hyperbolic triangle).
• Week 17: Klein disk model. Model on two-sheet hyperboloid.
• Week 18: Isometries in hyperbolic geometry: reflections as generators of the isometry group, classification of isometries, invariant sets (circles, horocycles and equidistant curves). Problem session at the second hour (solving problems in different models, computations in the upper half-plane and in the hyperboloid model).
• Week 19: Hyperbolic surfaces (examples of constructions: gluing from polygons, pants decompositions, quotients of the hyperbolic plane by discrete subgroups which act freely). Hyperbolic 3-space.
• Homeworks:

• There will be 2 sets of marked homework assignments per term in weeks 14 and 18.
• In addition, there will be weekly unmarked sets of exercises.
• Week 11: ---- Exercises ----
• Week 12: ---- Exercises ----
• Week 13: ---- Exercises ----
• Week 14: ---- Exercises ---- ---- Marked HW assignment. Due: Friday, February 15
• Week 15: ---- Exercises ----
• Week 16: ---- Exercises ----
• Week 17: ---- Exercises ----
• Week 18: ---- Exercises ---- ---- Marked HW assignment. Due: Friday, March 15
• Week 19: ---- Exercises ---- ---- Extra set: exercises on circles, horospheres and equidistant curves

Handouts: