My interests are centered about hyperbolic geometry. I am keen to pursue my further study in mathematics in the following fields:
- The geometry of $3$-dimensional hyperbolic and anti-de Sitter manifolds.
- The relations between $3$-dimensional geometry and Teichmüller theory.
- The discrete and polyhedral geometry in relation with constant curvature spaces.
Moreover, I am planning to learn the above said topics (or some parts of the topics) on my own.
Previously, I studied the first four chapters of "Fuchsian Groups" by S. Katok and some parts of "Riemann Surfaces" by S Donaldson. I found that mathematicians who do hyperbolic geometry are also interested about hyperbolic $3$-manifolds. There are many interested things about $3$-manifolds. I have found out the interesting one is "Kleinian Groups and Thurston's Work". Consider the hyperbolic $3$ space $\mathbb{H} ^3 = \{(x, y, z) \in \mathbb{R}^3 : z > 0\}$ endowed with the metric $ds^2 = \frac{dx^2 +dy^2+ dz^2}{z^2}$. The discrete subgroup of the isometry group of the hyperbolic $3$-space is called the Kleinian groups. Now one can construct all-right hyperbolic dodecahedron (a dodecahedron in hyperbolic space such that all faces are totally geodesic regular hyperbolic pentagons, and all dihedral angles are right angles). Now, gluing opposite faces of the dodecahedron by a three-fifths twist one can obtain a closed hyperbolic $3$-manifold (Seifert-Weber space). The group of face-pairing transformations generate a Kleinian group such that the quotient of hyperbolic $3$-space by this group is the Seifert-Weber space. In $3$ dimensions, there are $8$ geometries, called the eight Thurston's geometries. Not every $3$-manifold admits a geometry, but Thurston's geometrization conjecture (now Perelman's theorem) states that every topological $3$-manifold can be canonically decomposed along spheres and tori such that each resulting pieces are geometrizable. It is an analogue of the uniformization theorem for two-dimensional surfaces (more especially for Riemann surfaces).
I have written about the $3$-manifolds (above said) in layman's term. As I wrote earlier that I want to prepare myself in the above topics, namely 1), 2), 3), I am looking for a learning roadmap for 1), 2), and 3). And I want to know what are the prerequisites for learning the above said topics. Also, how much of Riemannian geometry and complex geometry need to cover to get into the said topics. Please advise me for a learning roadmap and references for the above said topics.
Please advise me in details.
Thanking in advance.