Title: Introduction to Hyperbolic Geometry.

Annotation: The lectures are intended for students majoring in mathematics. They can be divided into 3 parts:
1. Hyperbolic plane.
2. Hyperbolic space.
3. Volumes of hyperbolic polyhedra.

In the first part, we will talk about what two-dimensional geometries of constant curvature exist. We will consider the Poincaré model of hyperbolic geometry in the upper half-space. What do geodesics look like in this model. We will derive the main relations in a triangle.

In the second part, we will consider models of hyperbolic space in dimension 3 and higher. We will dwell on the isometry group of three dimensional Lobachevsky space and how it is related to fractional-linear mappings on the extended complex plane.

In the third part, we will consider an ideal tetrahedron and derive a formula for its volume in terms of the Lobachevsky function. We will derive formulas for the volumes of some other ideal polyhedra (pyramids, prisms, antiprisms). We will give examples of constructing three dimensional manifolds by gluing the faces of polyhedra. We will talk about the Schläfli differential equation and about different approaches to calculating hyperbolic volumes.