# Office of Academic AffairsIndian Institute of Science Education and Research Berhampur

Mathematics

MTH 520: Introduction to Hyperbolic Geometry (4)

Pre-requisites (Desirable): MTH 304, MTH 407

Learning Objectives:

Hyperbolic geometry is arguably the most important area in modern geometry and topology. This course is intended to expose the student to the foundational concepts in hyperbolic geometry, and is specially tailored to prepare the student for advance topics in geometric topology.

Course Contents:

The general Möbius group. The extended complex plane (or the Riemman sphere) C; The general Möbius group Mob(Ĉ); Identifying Mob+(Ĉ) with the matrix group PGL(2; C); Classification of elements of elements of Mob+(Ĉ); Reflections and the general Möbius group Mob(Ĉ); Conformality of elements in Mob(Ĉ).

The upper-half plane model H2. The upper half planeH2; The subgroup Mod(H2); Transitivity properties of Mob+(H2); Geometry of the action of Mob+(H2); The metric inH2; Element of arc-length inH2; Path metric inH2; The Poincaré metric dH onH2; Geodesics inH2; Identifying the group Mob+(H2) of isometries of (H2, dH) with PSL(2; R); Ultraparallel lines in H2.

The Poincaré disk model D. The Poincaré disk D; Transitioning fromH2to D via Mob+(H2); Element of arc-length and the metric dD in D; The Group Mob(D) of isometries of (D, dD); Centre, radii, and length of hyperbolic circles in D; Hyperbolic structures on holomorphic disks.

Properties of H2. Curvature of H2; Convex subsets of H2; Hyperbolic polygons; Area of a subset of H2; Gauss-Bonnet formula - area of a hyperbolic triangle; Applications of Gauss-Bonnet Formula: Area of reasonable hyperbolic polygons, existence of certain hyperbolic n-gons, hyperbolic dilations; Putting a hyperbolic structure on a surface using hyperbolic polygons; Hyperbolic trigonometry: triogometric identities, law of sines and cosines, Pythagorean theorem.

Non-planar models (if time permits). Hyperboloid model for the hyperbolic plane; Higher dimensional hyperbolic spaces.

Suggested Books:

• James W. Anderson, Hyperbolic Geometry (2nd Edition), Springer, 2005.
• Arlan Ramsay, Robert D. Richtmyer, Introduction to Hyperbolic Geometry, Springer, 1995.
• Harold E. Wolfe, Introduction to Non-Euclidean Geometry, Dover, 2012
• Alan F. Beardon, The geometry of discrete groups (Chapter 7), Springer, 1983.
• Svetlana Katok, Fuchsian Groups (Chapter 1), Chicago Lectures in Mathematics, 1992.
• John Stillwell, Geometry of surfaces (Chapter 4), Springer, 1992.