We briefly review some basic facts about the theory of Riemannian geometry and present more technical details in the Appendix. The reader can refer to (Spivak 1979, Lang 1999, Boothby 1986, do Carmo 1992, Pennec 2006) for more details.
A Riemannian manifold (M,m) is a smooth manifold M ⊂ RdM together with
an inner product m, wheredM is the dimension of M. We first introduce the tangent vector and tangent space atp∈ M. For a small scalarδ >0, letγ(t)be a differentiable map from(−δ, δ) toM passing through γ(0) = p. A tangent vector at pis defined as the derivative of the smooth curve γ(t) with respect to t evaluated at t = 0. The set of all tangent vectors at p forms the tangent space of M at p, denoted as TpM. The
TpMis equipped with an inner product mp, called a Riemannian metric, which varies
smoothly from point to point. IfMis complete, theexponential map atpis defined on the tangent spaceTpMbyExpMp (V) =γ(1; p, V), whereγ(1; p, V)is the geodesic with
γ(0; p, V) = p and γ0(0; p, V) = V. An open subset U of M containing p is a normal chart nearp if ExpMp is a diffeomorphism on an open neighborhood V of the origin in
TpMontoU withV such thattV ∈ V for0≤t≤1andV ∈ V. The inverse map is the
logarithmic mapat p, denoted by LogMp . Then, for q ∈ U, distM(p,q) =kLogMp (q)kp.
Theradius of injectivityofMatp, denoted by ρ∗(M,p), is the largestr >0such that ExpMp is a diffeomorphism on the open ball B(0, r)⊂TpMonto an open set inMnear
and then the logarithmic map Logp provides a local chart near p. If TpM is endowed
with an orthonormal basis, such a chart is called a normal chart and the coordinates are called normal coordinates.
A Lie group G is a group together with a smooth manifold structure such that
the operations of multiplication (a, b) 7→ ab and inversion a 7→ a−1 are smooth maps.
Throughout this work, we only consider finite dimensional Lie groups. Theexponential mapofGat its identity element, denoted bye, is defined asExpG(v) = γG(1;v)for any v ∈ TeG, where γG(·;v) : R → G is the unique one-parameter subgroup of G whose
tangent vector ate is equal to v. For a ∈G, the exponential map of G at a is defined
by ExpGa ◦ La∗ = La ◦ExpGe, where La is the left multiplication by a and La∗ is its tangent map. Many common geometric transformations of Euclidean spaces including rotations, translations, dilations, and affine transformations onRd form Lie groups. In
general, Lie groups can be used to describe transformations of smooth manifolds. A RSS is a connected Riemannian manifoldMwith the property that at each point, the mapping that reverses geodesics through that point is an isometry. Examples of RSS’s include Euclidean spaces,Rk, spheres,Sk, projective spaces,P Rk, and hyperbolic
spaces, Hk, each with their standard Riemannian metrics. Symmetric spaces arise
naturally from Lie group actions on manifolds. Given a smooth manifold M and a Lie group G, a smooth group action of G on M is a smooth mapping G× M → M, (a,p) 7→ a·p such that e·p = p and (aa0)·p = a·(a0 ·p) for all a, a0 ∈ G and all p∈ M. The group action should be interpreted as a group of transformations of the manifold M, namely, {La}a∈G such that La :M → M, La(p) = a·p for p∈ M and
a ∈ G. The La is a smooth transformation on M and its inverse is denoted by La−1. Givenp∈ M, letιp denote the action ofGon the pointp∈ Msuch thatιp :G→ M,
ιp(a) = a·p = La(p) for all a ∈ G. Thus, ιp is a smooth map from G into M. The
of M. Specifically, two points p,p0 ∈ Mare equivalent if there exists a ∈Gsuch that a·p = p0. If M consists of a single orbit, the group action is transitive or G acts
transitively on M, and we call Mas a homogeneous space. The isotropy subgroup of a pointp∈ M is defined as Gp ={a∈G|a·p = p}.
When a Lie groupGacts smoothly on a smooth manifoldM, for anyp∈ M, there is a natural bijection from the orbitG(p)onto the quotient manifold given by the smooth mappinga·p7→aGp such thatG(p)∼=G/Gp, where∼=denotes the bijection. LetGbe
a connected group of isometries of the RSSMsuch that distM(p,p0) =distM(a·p, a·p0), for all p,p0 ∈ M and alla ∈G. For any p∈ M, the RSS Mcan always be viewed as a homogeneous space, M ∼=G/Gp, and the isotropy subgroup Gp is compact.
From now on, we will assume that the manifoldMis a RSS andM=G/Gp withG
being a Lie group of isometries acting transitively onM. Geodesics onMare computed through the action ofGonM. Due to the transitive action of the groupGof isometries
on M, it suffices to consider only the geodesic starting at the base point p. For any pointy∈ M, geodesics starting fromy are of the forma·γ(·), where γ(·)is a geodesic starting from p, γ(0) = p and y =a·p for some a ∈ G. Due to the local uniqueness
of geodesics, if y =a0 ·p for some other a0 ∈ G, then a·γ(·) = a0 ·γ(·). Geodesics on Mstarting fromp are the images of the action of a 1-parameter subgroup ofGacting
on the base point p. In other words, for any geodesicγ onM, γ(·) :R → M, starting fromp, there exists a 1-parameter subgroup c(·) :R → G such that γ(t) = c(t)·p for allt∈R.