Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

Chapter 1
Tools of Riemannian Geometry
1.1 Settings
Definition 1.1.1. A topological space is a set X together with a collection O of subsets of X
satisfying the following properties
• ∅, X ∈ O
• the intersection of any finite collection of sets in O is also in O,
• the union of any collection of sets in O is also in O.
The sets in O are called open.
Definition 1.1.2. A topological space is Hausdorff if for any two distinct points x1, x2 ∈ X there
exists open sets O1, O2 ∈ O such that x1 ∈ O1, x2 ∈ O2 and O1 ∩ O2 = ∅.
The Hausdorff axiom is essential for the uniqueness of the limit of convergent sequences.
Definition 1.1.3. A map between two topological spaces is called continuos if the inverse image
of any open set is open. A bijective map which is continuous in both directions is called
homeomorphism.
Definition 1.1.4. A n-dimensional manifold is a Hausdorff topological space M such that for each
x ∈ M there is a neighborhood U of x that is homeomorphic to an open subset O of R n . Such a
homeomorphism ϕ : U → O is called coordinate chart.
The important point here is that locally the topology of a manifold is the same as that of R n .
Definition 1.1.5. Consider a family (Ui)i∈I of open sets such that M =
i∈I Ui. An atlas is the
family {Ui, ϕi}i∈I of charts.
Remark 1.1.1. Each ϕ −1
i (·) is referred to as parametrization and the set ϕi(Ui) is called parameter
domain.
Definition 1.1.6. A n-dimensional manifold M is C r differentiable if, for Ui ∩ Uj = ∅, the
composition
ϕj ◦ ϕ −1
i
: ϕi(Ui ∩ Uj) → ϕj(Ui ∩ Uj)