Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

3 Tools of Riemannian Geometry
For a proof see, for example, Kühnel (24) Theorem 5.6. Therefore for every tangent vector X(p) ∈
TpM one has
n
X(p) =
i=1
ξ i p ∂i(p), (1.3)
where ξ i p is the directional derivative of the coordinate function x i in the direction X(p). For a
function f : M → R, f ∈ C 1 we have
X(p)f =
n
i=1
ξ i p ∂i(p)f.
Definition 1.1.10. A differentiable vector field X on a n-dimensional manifold M is an association
p ∈ U ⊂ M → X(p) ∈ TpM such that in every chart ϕ : U → R n , with coordinates (x 1 , . . . , x n ), the
coefficients ξ i p : U → R in the representation (1.3) valid at the each point p ∈ U are differentiable
functions.
Common notations for the vector field and the directional derivative of a function f : M → R,
f ∈ C 1 in the direction of the vector field are
X =
n
ξ i ∂i, Xf =
i=1
n
ξ i ∂if.
The interpretation of X as a derivative operator permits us to consider the iterates of X. If
X and Y are two differentiable vector fields on M and f : M → R, f ∈ C 2 , we can consider the
functions X(Y f) and Y (Xf).
Proposition 1.1.1. Let X and Y be two differentiable vector fields on a C ∞ manifold M. Then
there exists a unique vector filed [X, Y ], which is referred to as the Lie bracket of X and Y , such
that for all f : M → R, f ∈ C 2 ,
i=1
[X, Y ]f = X(Y f) − Y (Xf).
For a proof see, for example, Do Carmo (14) Lemma 5.2.
We introduce now a metric structure on M.
Definition 1.1.11. An inner product on a vector space V over a field F is a bilinear form
such that is
• symmetric: 〈v, w〉 = 〈w, v〉
• positive definite: 〈v, v〉 > 0 for all v = 0.
(v, w) ∈ V × V → 〈v, w〉 ∈ F,
Definition 1.1.12. Given a C ∞ manifold M, define a Riemannian metric g on M to be a mapping
that associates with each p ∈ M an inner product
gp : MpT × MpT → R