Covariant Derivatives on Homogeneous Spaces -- Horizontal Lifts and Parallel Transport

We consider invariant covariant derivatives on reductive homogeneous spaces corresponding to the well-known invariant affine connections. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields on the Lie group. This point of view allows for a characterization of parallel vector fields along curves. Moreover, metric invariant covariant derivatives on a reductive homogeneous space equipped with an invariant pseudo-Riemannian metric are characterized. As a by-product, a new proof for the existence of invariant covariant derivatives on reductive homogeneous spaces and their the one-to-one correspondence to certain bilinear maps is obtained.


Introduction
Reductive homogeneous spaces play a role in a wide range of applications from mathematical physics to an engineering context. Without going into details, geodesics and parallel transport are certainly of interest. These notions can be defined with respect to invariant covariant derivatives which correspond to the well-known invariant affine connections from the literature. In fact, the existence of invariant affine connections on a reductive homogeneous space G/H with a fixed reductive decomposition g = h ⊕ m and their oneto-one-correspondence to Ad(H)-invariant bilinear maps m × m → m were proven in [13].
The initial motivation for this text was to derive a characterization of parallel vector fields along curves generalizing [7,Lem. 1] to an arbitrary reductive homogeneous space equipped with some invariant covariant derivative. In order to obtain such a characterization, given in Corollary 4.27 below, we express an arbitrary invariant covariant derivative on G/H associated to an Ad(H)-invariant bilinear map m×m → m in terms of horizontally lifted vector fields on G. This expression generalizes formulas for the Levi-Civita covariant derivative on G/H in terms of horizontally lifted vector fields from the literature, where G is equipped with a bi-invariant metric and G/H is endowed with a pseudo-Riemannian metric such that G → G/H is a pseudo-Riemannian submersion. Indeed, in the proof of [7, Lem. 1], a formula for the Levi-Civita covariant derivative on a pseudo-Riemannian symmetric space in terms of horizontally lifted vector fields is obtained. Moreover, a formula for the Levi-Civita covariant derivative in terms of horizontal vector fields is derived in [15,Sec. 4.2] for certain homogeneous spaces of compact Lie groups equipped with bi-invariant metrics. Here we also mention the recent work [18], where similar questions are independently discussed in the context of spray geometry.
We now give an overview of this text. We start with introducing some notations in Section 2. Moreover, in Section 3, we recall some facts on reductive homogeneous spaces and discuss the principal connections defined by reductive decompositions. After this preparation, we come to Section 4, where invariant covariant derivatives are investigated in detail. In Subsection 4.1, we show that an invariant covariant derivative is uniquely determined by evaluating it on certain fundamental vector fields of the left action G × G/H ∋ (g, g ′ · H) → (gg ′ ) · H ∈ G/H. Afterwards, we express an invariant covariant derivative ∇ α corresponding to an Ad(H)-invariant bilinear map α : m × m → m in terms of horizontally lifted vector fields on G as follows. For two vector fields X and Y on G/H whose horizontal lifts are the vector fields on G denoted by X and Y , respectively, we express the horizontal lift ∇ α X Y of ∇ α X Y in terms of X and Y . The exact expression for ∇ α X Y is obtained in Theorem 4.15. As a by-product, a new proof for the existence of invariant covariant derivatives associated to Ad(H)-invariant bilinear maps m × m → m is obtained. Moreover, the formula from Theorem 4.15 is used to derive the curvature of ∇ α in Subsection 4.2 In addition, we characterize invariant metric covariant derivatives if G/H is equipped with an invariant pseudo-Riemannian metric. In Subsection 4.4, we turn our attention to vector fields along curves. In particular, the expression of ∇ α in terms of horizontally lifted vector fields from Theorem 4.15 allows for characterizing parallel vector fields along curves on G/H in terms of an ODE on m. In addition, we obtain a geodesic equation for the reductive homogeneous space G/H equipped with an invariant covariant derivative. If this geodesic equation is specialized to a Lie group endowed with some left-invariant metric, the well-known geodesic equation from [1, Ap. 2] is obtained. Finally, we discuss the canonical invariant covariant derivatives of first and second kind which correspond to the canonical affine connections of first and second kind from [13,Sec. 10].

Notations and Terminology
We start with introducing the notation and terminology that is used throughout this text.
Notation 2. 1 We follow the convention in [14,Chap. 2]. A scalar product is defined as a non-degenerated symmetric bilinear form. An inner product is a positive definite symmetric bilinear form.
Next we introduce some notations concerning differential geometry. Let M be a smooth Let E → M be a vector bundle over M with typical fiber V . The smooth sections of E are denoted by Γ ∞ (E). We write End(E) ∼ = E * ⊗ E for the endomorphism bundle of E. Moreover, we denote by E ⊗k , S k E and Λ k E the k-th tensor power, the k-th symmetrized tensor power and the k-th anti-symmetrized tensor power of E. If T ∈ Γ ∞ (T * M ) ⊗k ⊗ (T M ) ⊗ℓ is a tensor field on M and X ∈ Γ ∞ (T M ) is a vector field on, we write L X T for the Lie derivative. The pull-back of a smooth function x : N → Ê by the smooth map f : M → N is denoted by f * x = x • f : M → Ê. More generally, if ω ∈ Γ ∞ Λ k (T * N ) ⊗ V is a differential form taking values in a finite dimensional Ê-vector space V , its pull-back by f is denoted by f * ω.
Concerning the regularity of curves on manifolds, we use the following convention.

Background on Reductive Homogeneous Spaces
In this section, we introduce some more notations and recall some well-known facts concerning Lie groups and reductive homogeneous spaces. Moreover, the principal connection on the H-principal fiber bundle G → G/H obtained by a reductive decomposition is discussed in detail.

Lie groups
We start with introducing some notations and well-known facts concerning Lie groups and Lie algebras. Let G be a Lie group and denote its Lie algebra by g. The identity of G is usually denoted by e. We write ℓ g : G → G, h → ℓ g (h) = gh (3.1) for the left translation by g ∈ G and the right translation by g ∈ G is denoted by The conjugation by an element g ∈ G is given by and the adjoint representation of G is defined as Moreover, we denote the adjoint representation of g by respectively, holds. For ξ ∈ g, we denote by ξ L ∈ Γ ∞ (T G) and ξ R ∈ Γ ∞ (T G) the corresponding left and right-invariant vector fields, respectively, which are given by ξ L (g) = T e ℓ g ξ and ξ R (g) = T e r g ξ, g ∈ G. (3.7) We write exp : g → G. (3.8) for the exponential map of G.

Reductive Homogeneous Spaces
Next we recall some well-known facts on reductive homogeneous spaces and introduce the notation that is used throughout this text. We refer to [5,Sec. 23.4] or [14,Chap. 11] for details. Let G be a Lie group and let g be its Lie algebra. Moreover, let H ⊆ G a closed subgroup whose Lie algebra is denoted by h ⊆ g. We consider the homogeneous space G/H. Then is a smooth action of G on G/H from the left, where g · H ∈ G/H denotes the coset defined by g ∈ G. Borrowing the notation from [5, p. 676], for fixed g ∈ G, the associated diffeomorphism is denote by In addition, we write for the canonical projection. Since reductive homogeneous spaces play a central role in this text, we recall their definition from [5,Def. 23.8], see also [13,Sec. 7] or [14,Def. 21,Chap. 11].
Definition 3.1 Let G be a Lie group and g be its Lie algebra. Moreover, let H ⊆ G be a closed subgroup and denote its Lie algebra by h ⊆ g. Then the homogeneous space G/H is called reductive if there exists a subspace m ⊆ g such that g = h ⊕ m is fulfilled and holds for all h ∈ H.
Following [5,Prop. 23.22], we recall a well-known property of the isotropy representation of a reductive homogeneous space. This is the next lemma.

Lemma 3.2
The isotropy representation of a reductive homogeneous space G/H with re- is equivalent to the representation i.e.
is fulfilled for all h ∈ H. Notation 3.3 Let g = h ⊕ m be a reductive decomposition of g. Then the projection onto m whose kernel is given by h is denoted by pr m : g → m. We write pr h : g → h for the projection whose kernel is given by m. Moreover, we write for ξ ∈ g ξ m = pr m (ξ) and ξ h = pr h (ξ).
The following lemma can be considered as a generalization of [5,Prop. 23.29 (1)-(2)] to pseudo-Riemannian metrics and Lie groups which are not necessarily connected.
Lemma 3.6 Let G be a Lie group and denote by g its Lie algebra. Moreover, let G be equipped with a bi-invariant metric and let ·, · : g × g → Ê be the corresponding Ad(G)invariant scalar product. Moreover, let H ⊆ G be a closed subgroup such that its Lie algebra h ⊆ g is non-degenerated with respect to ·, · . Then G/H is a reductive homogeneous space with reductive decomposition g = h ⊕ m, where m = h ⊥ is the orthogonal complement of h with respect to ·, · . Moreover, if G/H is equipped with the invariant metric corresponding to the scalar product on m that is obtained by restricting ·, · to m, the reductive homogeneous space G/H is naturally reductive.
Proof: The claim can be proven analogously to the proof of [5,Prop. 23.29 (1)-(2)] by taking the assumption h ⊕ h ⊥ = h ⊕ m = g into account.
Remark 3.7 Inspired by the terminology in [5, Sec. 23.6, p. 710], we refer to the naturally reductive homogeneous spaces from Lemma 3.6 as normal naturally reductive homogeneous spaces.
We end this subsection with considering another special class of reductive homogeneous spaces. To this end, we state the following definition which can be found in [6, p. 209]. Inspired by the terminology used in [5,Def. 23.13], we refer to the triple (G, H, σ) as symmetric pair, as well, where (G, H) is a symmetric pair with respect to the involutive automorphism σ : G → G. These symmetric pairs lead to reductive homogeneous spaces which are called symmetric homogeneous spaces if a certain "canonical" reductive decomposition is chosen, see e.g. [13,Sec. 14]. Note that the definition in [13,Sec. 14] does not require an invariant pseudo-Riemannian metric on G/H. The next lemma, see e.g. [13,Sec. 14], shows that a symmetric homogeneous space is a reductive homogeneous space with respect to the so-called canonical reductive decomposition. Here we also refer to [5,Prop. 23.33] for a proof. (3.20) Then g = h⊕m is a reductive decomposition of g turning G/H into a reductive homogeneous space. Moreover, the inclusion is fulfilled.
Definition 3.10 Let (G, H, σ) be a symmetric pair. Then the reductive decomposition g = h ⊕ m from Lemma 3.9 is called canonical reductive decomposition. Moreover, the reductive homogeneous space G/H with the reductive decomposition g = h ⊕ m is called symmetric homogeneous space.
Remark 3.11 Let (G, H, σ) be symmetric pair and let G/H be the associated symmetric homogeneous space with canonical reductive decomposition g = h ⊕ m. Let G/H be equipped with an invariant pseudo-Riemannian metric and let ·, · : m × m → Ê be the associated Ad(H)-invariant scalar product. Then G/H is a naturally reductive homogeneous space since [m, m] ⊆ h implies that the condition on the scalar product ·, · from Definition 3.5 is always satisfied. In the sequel, we refer to symmetric homogeneous spaces equipped with an invariant pseudo-Riemannian metric as pseudo-Riemannian symmetric homogeneous space or pseudo-Riemannian symmetric spaces, for short.

Reductive Decompositions and Principal Connections
In this section, we consider G as a H-principal fiber bundle over G/H and discuss certain principal connections on pr : G → G/H. For general properties of principal fiber bundles and connections, we refer to [ Let G be a Lie group and H ⊆ G be a closed subgroup. It is well-known that pr : G → G/H is a H-principle fiber bundle, see e.g. [12,Sec. 18.15], where the base is the homogeneous space G/H. The H-principal action on G is denoted by if not indicated otherwise. We now assume that G/H is a reductive homogeneous space and the reductive decomposition g = h ⊕ m is fixed. This reductive decomposition can be used to obtain a principal connection on pr : G → G/H, see [9,Thm. 11.1]. Although this fact is well-known, we provide a detailed proof in order to keep this text more selfcontained. To this end, we recall a well-known fiber-wise expression for the vertical bundle of pr : G → G/H which follows for example from [ The next proposition provides explicit formulas for the principal connection and the associated principal connection one-form on G → G/H defined by a reductive decomposition.
Proposition 3.12 Consider pr : G → G/H as a H-principal fiber bundle, where G/H is a reductive homogeneous space with a fixed reductive decomposition g = h ⊕ m and define Hor(G) ⊆ T G fiber-wise by (3.24) Then Hor(G) is a subbundle of T G defining a horizontal bundle on T G, i.e. a complement of the vertical bundle Ver(G) = ker(T pr) ⊆ T G which yields a principal connection on pr : G → G/H. This principal connection P ∈ Γ ∞ End(T G) corresponding to Hor(G) is given by The corresponding connection one-form ω ∈ Γ ∞ (T * G) ⊗ h reads for g ∈ G and v g ∈ T g G.
Proof: Although, this statement is well-known, see e.g. [9, Thm. 11.1], we provide a proof, nevertheless. Indeed, Hor(G) is a complement of the vertical bundle Ver(G) = ker(T pr) ⊆ T G due to g = h ⊕ m implying T G = Ver(G) ⊕ Hor(G) as desired. Moreover, P defined by (3.25) is clearly a smooth endomorphism of the vector bundle T G → G, i.e. P ∈ Γ ∞ End(T G) . In addition, P 2 = P is obviously fulfilled. Moreover, one has im(P) = ker(T pr) = Ver(G) and ker(P) = Hor(G) showing that P is the connection corresponding to the horizontal bundle Hor(G). We now show that ω is the connection one-form corresponding to P by using the correspondence from [12,Sec. 19.1]. Let η ∈ h and denote by η G the corresponding fundamental vector field, i.e. we have for g ∈ G η G (g) = d dt g ⊳ exp(tη) t=0 = d dt ℓ g exp(tη) t=0 = T e ℓ g η.
By this notation, one obtains for v g ∈ T g G Moreover, we have for all η ∈ h proving that ω ∈ Γ ∞ (T * G) ⊗ h is the connection one-from corresponding to P ∈ Γ ∞ End(T G) .
It remains to show that P is a principal connection. By [12,Sec. 19.1] this is equivalent to showing that ω has the equivariance property for all h ∈ H, g ∈ G and v g ∈ T g G, where (· ⊳ h) * ω denotes the pull-back of ω by (· ⊳ h) : P ∋ p → p ⊳ h ∈ P . Since g = h ⊕ m is a reductive decomposition, we obtain for h ∈ H and ξ ∈ g (3.27) Using (3.27) and the chain-rule, we compute for h ∈ H, g ∈ G and v g ∈ T g G Hence ω is the connection one-form corresponding to the principal P.
By [5,Prop. 23.23], adapted to the pseudo-Riemannian case, we obtain the following remark concerning pseudo-Riemannian reductive homogeneous spaces.
Remark 3.13 Let G/H be a reductive homogeneous space with reductive decomposition g = h ⊕ m endowed with an invariant pseudo-Riemannian metric corresponding to the Ad(H)-invariant scalar product ·, · : m × m → Ê. By [5,Prop. 23.23], which clearly extends to the pseudo-Riemannian case, the scalar product ·, · on m can be extended to a scalar product ·, · g on g such that m = h ⊥ is fulfilled. Then pr : G → G/H becomes a pseudo-Riemannian submersion by [5,Prop. 23.23], where G is equipped with the leftinvariant metric defined by ·, · g . Obviously, the horizontal bundle defined by Hor(G) = Ver(G) ⊥ yields the connection on G which coincides with the principal connection from Proposition 3.12 defined by the reductive decomposition g = h ⊕ m.

Invariant Covariant Derivatives
In this section, we consider the invariant covariant derivatives on a reductive homogenoeus space G/H that correspond to the invariant affine connections investigated in [13]. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields yielding another proof for their existence. In particular, this expression is used to characterize parallel vector fields along curves in terms of an ODE on m.
Throughout this subsection, we use the following notation.
Notation 4.1 If not indicated otherwise, we denote by G/H a reductive homogeneous space with a fixed reductive decomposition g = h ⊕ m.

Invariant Covariant Derivatives
We start with introducing the notion of an invariant covariant derivative on a reductive homogeneous space. In view of the one-to-one correspondence of covariant derivatives and affine connections, see Remark 4.3 below, the next definition can be seen as a reformulation of [ holds for all g ∈ G and X, Y ∈ Γ ∞ T (G/H) , where (τ g ) * X denotes the well-known pushforward of X by the diffeomorphism τ g : Obviously, for a fixed g ∈ G the push-forward In the next remark, we relate the notion of affine connections from [13] to covariant derivatives.

Remark 4.3 Let M be a manifold and denote by End
such that holds for all X 1 , X 2 , X, Y ∈ Γ ∞ (T M ). As pointed out in [2,Sec. 4.5], an affine connection t : Obviously, the converse is also true. Given a covariant derivative ∇ on T M , Equation (4.6) yields an affine connection.
In the sequel, we discuss the invariant covariant derivatives on G/H corresponding to the invariant affine connections on G/H from [13,Thm. 8.1]. This correspondence is made precise in Proposition 4.18, below.
We first recall the notion of an Ad(H)-invariant bilinear map from [13,Sec. 8].
holds for all X, Y ∈ m and h ∈ H. More generally, for ℓ ∈ AE, we call a ℓ-linear map holds for all X 1 , . . . , X ℓ ∈ m and h ∈ H.
Remark 4.5 As we have already pointed out in the introduction, the one-to-one correspondence between invariant affine connections and Ad(H)-invariant bilinear maps m × m → m is well-known by [13,Thm. 8.1]. Nevertheless, the discussion in this text differs from the discussion in [13]. Inspired by [5,Sec. 23.6], we consider invariant covariant derivatives evaluated at the fundamental vector fields of the action τ : G × G/H → G/H at the point pr(e) which already determines them uniquely. Moreover, we express invariant covariant derivatives on G/H in terms of horizontally lifted vector fields on G. Beside yielding another proof for the existence of an invariant covariant derivative associated with an Ad(H)-invariant bilinear map m × m → m, this point of view allows in particular for an easy characterization of parallel vector fields, see Subsection 4.4 below.

Invariant Covariant Derivatives evaluated on Fundamental Vector Fields
Before we continue with considering invariant covariant derivatives, we take a closer look on the fundamental vector fields on G/H associated with the action τ : In the next lemma, we state some properties of X G/H . Note that its third claim is well-known.
Then the following assertions are fulfilled: 1. The horizontal lift of X G/H is given by for all g ∈ G.
2. Let Y ∈ m and define the smooth functions y j : where j ∈ {1, . . . , N }. Then one has for all g ∈ G and pr(k) ∈ G/H.

Proof:
We first show Claim 1. To this end, we compute for g ∈ G showing that X G/H is pr-related to the right-invariant vector field X R . Next we express X R in terms of left-invariant vector fields. Let g ∈ G. We now compute (4.14) Let P be the principal connection from Proposition 3.12. Then the horizontal lift of X G/H is given by Using (4.14) and Next we show Claim 2. The curve γ : Therefore we compute, again by pr Although a proof of Claim 3 can be found for example in [5,Prop. 23.20], following this reference, we repeat it here for the reader's convenience. We compute for g, k ∈ G where (4.3) is used in the first equality and we also exploited Conj g • exp = exp • Ad g . This yields the desired result.
It is well-known that there is a one-to-one correspondence between Ad(H)-invariant tensors on m and invariant tensor fields on G/H, see e.g. [14,Chap. 11,p. 312]. In the sequel, we need the following lemma which can be regarded as a special case of this assertion. In order to keep this text as self-contained as possible, we include proof which is inspired by the proof of [5,Prop. 23.23] and [14,Chap. 11,Prop. 22]. Lemma 4.7 Let G/H be a reductive homogeneous space with reductive decomposition g = h ⊕ m. There is a one-to-one-correspondence between Ad(H)-invariant ℓ-linear maps for all X 1 , . . . , X ℓ ∈ Γ ∞ (T M ) and g ∈ G by requiring for all X 1 , . . . , X ℓ ∈ m.
Proof: We use ideas that can be found in [5,Prop. 23.23], see also [14,Chap. 11,Prop. 22]. In this proof, we write o = pr(e) = e · H ∈ G/H for the coset defined by e ∈ G. Let We first show that this yields a well-defined expression. Let k ∈ G be another element with pr(g) = p = pr(k), i.e. there exists a h ∈ H with g = kh and therefore k −1 = hg −1 as well as k = gh −1 is fulfilled. Using the definition of D in (4.18) and (4.19), we compute where the fourth equality follows by a calculation similar to (4.17) exploiting the Ad(H)invariance of d : m ℓ → m. It remains to proof that D has the desired invariance property. To this end, let X 1 , . . . , X ℓ ∈ Γ ∞ T (G/H) be vector fields and let g ∈ G. Suppressing the "foot points" of the tangent maps, we compute by the definition of D for q = pr(k) ∈ G/H represented by some k ∈ G .
In the remainder part of this subsection, we investigate invariant covariant derivatives and their relation to Ad(H)-invariant bilinear maps m × m → m. We first show that an invariant covariant derivative on G/H yields an Ad(H)-invariant bilinear map by evaluating it on fundamental vector fields and considering its value at pr(e) ∈ G/H. This is motivated by the discussion in [5,Sec. 23.6].
Before we proceed, we point out that the right-hand side of (4.20) in the next lemma is chosen such that it coincides with the expression from Definition 4.16, below.
Moreover, by exploiting that T e pr m : m → T pr(e) (G/H) is a linear isomorphism, Claim 1 is equivalent to the assertion that is an Ad(H)-invariant bilinear map. The map β is bilinear since the covariant derivative where the last equality holds by (4.21). Obviously, (4.22) is equivalent to for all h ∈ H and X, Y ∈ m. Hence Claim 1 is proven.

Invariant Covariant Derivatives in Terms Horizontal Lifts
2. Let f : G → Ê be smooth and let X ∈ m. Moreover, let g ∈ G. Denoting by Proof: The first claim is obvious. It remains to prove the second claim. To this end, we compute for g, k ∈ G where exploited that X L ∈ Γ ∞ Hor(G) is a left-invariant vector field. This yields the desired result.
. . , N }. Using this notation and Einstein summation convention, as usual, we set Then (4.24) defines a map ∇ Hor,α : for all f ∈ C ∞ (G) and X, Y ∈ Γ ∞ Hor(G) . Moreover, ∇ Hor,α has the following properties: 1. For each g ∈ G, the map ∇ Hor,α is invariant under ℓ g : G → G in the sense that holds.
Horizontal lifts are compatible with push-forwards in the following sense.
Proof: Let g ∈ G. We have pr •ℓ g = τ g • pr implying T pr •T ℓ g = T τ g • T pr. Using this equality as well as T pr •X = X • pr we compute Since (ℓ g ) * X ∈ Γ ∞ Hor(G) is horizontal and T pr • (ℓ g ) * X = (τ g ) * X • pr holds by (4.31), we obtain (τ g ) * X = (ℓ g ) * X as desired. (4.32) Then X is the horizontal lift of X ∈ Γ ∞ T (G/H) given by holds for all g ∈ G and h ∈ H.
Proof: We first assume that X = x i A L i is the horizontal lift of the vector field X ∈ Γ ∞ T (G/H) . Then (4.33) holds. Using pr(gh) = pr(g) for all g ∈ G and h ∈ H, we can rewrite (4.33) equivalently as Since T g pr : Hor(G) g → T pr(g) (G/H) is a linear isomorphism for each g ∈ G, Equation (4.35) is equivalent to the left-hand side of (4.34). Applying the linear isomorphism (T e ℓ g ) −1 : Hor(G) g → m to both sides of this equality shows the equivalence to right-hand side of (4.34).
Conversely, assuming that the functions where the coset pr(g) = g · H ∈ G/H is represented by g ∈ G. Then the computation in (4.35) shows that X : G/H → T (G/H) is well-defined, i.e. we have for all g ∈ G and h ∈ H X(pr(g)) = T g pr •X(g) = T gh pr •X(gh) = X(pr(gh)).

(4.36)
Then X • pr = T pr •X holds by construction. Since pr : G → G/H is a surjective submersion and T pr •X : G → T (G/H) is smooth, the map X : G/H → T (G/H) is smooth by [10,Thm. 4.29]. Clearly, for pr(g) ∈ G/H, one has X(pr(g)) ∈ T pr(g) (G/H). Hence X ∈ Γ ∞ T (G/H) is a smooth vector field on G/H. Obviously, its horizontal lift is given by X.
holds. In particular, L X y j A L j ∈ Γ ∞ Hor(G) is the horizontal lift of the vector field X ∈ Γ ∞ T (G/H) given by X • pr = T pr • (L X y j )A L j .
for all g ∈ G and h ∈ H by Lemma 4.13. Let j ∈ {1, . . . , N }. Applying A j ∈ m * to (4.38) for all g ∈ G and h ∈ H. Next we define the curves c 1 : Ê ∋ t → g exp tx i (g)A i ∈ G and c 2 : holds and analogously one obtainṡ Expressing y j : G → Ê by (4.39) and using the definition of c 1 and c 2 , we compute for showing ( defines an invariant covariant derivative ∇ α : is fulfilled. In addition, ∇ α is the unique invariant covariant derivative on G/H satisfying (4.42).
Proof: We define the covariant derivative ∇ α on G/H by where ∇ Hor,α is given by Lemma 4.10. We first show that this definition yields a well-defined expression, i.e. ∇ α X Y ) • pr(g) = ∇ α X Y • pr(gh) holds for g ∈ G and h ∈ H. To this end, we calculate by exploiting Lemma 4.13 and Lemma 4.14 as well as the Ad(H)-invariance Hence (4.43) yields a well-defined vector field on G/H by Lemma 4.13.
Next we show that ∇ α yields a covariant derivative on G/H. Let f : G/H → Ê be smooth. By f X = pr * (f )X and the properties of ∇ Hor,α from (4.25) in Lemma 4.10, we obtain [10,Prop. 8.16] since X and X are pr-related. Moreover, we have Lemma 4.10. Hence ∇ α is indeed a covariant derivative. In addition, ∇ α is invariant. Indeed, by Lemma 4.10, Claim 1 and Lemma 4.12, one has Next let X, Y ∈ m and let {A 1 , . . . , A N } ⊆ m * be the dual basis of {A 1 , . . . , A N }. By Lemma 4.6, Claim 1, we have Y G/H = y j A L j with y j : G ∋ g → y j (g) = A j (Ad g −1 (Y )) ∈ Ê for j ∈ {1, . . . , N }. Thus we obtain by Lemma 4.6, Claim 2 ∇ Hor,α is called the invariant covariant derivative associated with α or corresponding to α.
where M is denoted by N in [13,Sec. 7]. Moreover, it is assumed that V is chosen such that the restriction of the canonical projection pr M : M → G/H is a diffeomorphism onto its image denoted by M * = pr(M ). It is pointed out in [13,Sec. 7] that the existence of such a chart is well-known referring to [3,Chap. IV,§V]. In addition to the assumptions from [13,Sec. 7], we assume that T e M = m holds. Clearly, a chart (V, x) of G centered at e ∈ G with the properties listed above can be constructed by exploiting that the map , we now define for X ∈ m the vector field X * ∈ Γ ∞ T M * by X * (pr(g)) = X * (τ g (pr(e))) = T pr(e) τ g (T e pr X), pr(g) ∈ M * , g ∈ M, (4.48) where we exploit that pr M : M → M * is a diffeomorphism. We now relate ∇ α to t α which is uniquely determined by t α (Y * )(X * ) pr(e) = T e pr α(X, Y ) , X, Y ∈ m (4.49) according to [13,Thm. 8.1], see in particular [13,Eq. (8.1)]. To this end, we rewrite (4.48) as X * (pr(g)) = T pr(e) τ g • (T e pr X) Analogously, one defines for Y ∈ m the vector field Y * on M * whose horizontal lift Clearly, the unique smooth functions  where we used (4.49) in the last equality. Moreover, ∇ α is the unique invariant covariant derivative on G/H satisfying (4.51). Indeed, let ∇ β be the invariant covariant derivative associated with the Ad(H)-invariant bilinear map β : yields β = α implying ∇ α = ∇ β . In addition, t α is uniquely determined by (4.49). Hence ∇ α and t α are both uniquely determined by (4.51). Thus (4.51) implies ∇ α X Y = t α (Y )(X) for all X, Y ∈ Γ ∞ T (G/H) as desired.

Torsion and Curvature
Next we consider the torsion of an invariant covariant derivative. This is the next lemma whose result coincides with [13, Eq. (9.2)].

Lemma 4.19
Let ∇ α be the invariant covariant derivative on G/H associated to the Ad(H)-invariant bilinear map α : m × m → m. The torsion of ∇ α is the G-invariant tensor field Tor α ∈ Γ ∞ Λ 2 (T * (G/H)) ⊗ T (G/H) defined by for all X, Y ∈ m Proof: We first note that (τ g ) * [X G/H , Y G/H ] = [(τ g ) * X G/H , (τ g ) * Y G/H ] holds all for g ∈ G, see e.g. [10,Cor. 8.31]. This identity and the invariance of ∇ α yields that Tor α is G-invariant. Thus Tor α corresponds to an Ad(H)-invariant bilinear map m × m → m by Lemma 4.7. In order to determine this bilinear map, writing pr(e) = o, we compute for all X, Y ∈ m, where we exploited that g ∋ X → X G/H ∈ Γ ∞ T (G/H) is an antimorphism of Lie algebras, see e.g. [12,Sec. 6.2].
Moreover, one can compute the curvature of ∇ α given by by using the expression for ∇ α from Theorem 4.15. This is the next proposition which yields an alternative derivation for the curvature obtained in [13, Eq. (9.6)].
for all X, Y, Z ∈ m.
Proof: Obviously, the curvature R α fulfills for all vector fields X, Y, Z ∈ Γ ∞ T (G/H) by the invariance of ∇ α . Hence R α is uniquely for all g ∈ G and i, j, k ∈ {1, . . . , N }. Using this notation, we obtain by Theorem 4.15 where the functions a ℓ : G → Ê for ℓ ∈ {1, . . . , N } are given by In particular, evaluating a ℓ : G → Ê at g = e yields by Lemma 4.6, Claim 2 (4.56) Moreover, we obtain by Theorem 4.15 and (4.56) (4.57) In order to obtain a more explicit expression for (4.57), we consider the first summand on the right-hand side. Recalling that a ℓ is given by (4.55) one obtains by the Leibniz rule (4.58) We now take a closer look at (4.58) evaluated at g = e. We obtain for second summand of its right-hand side by Lemma 4.6, Claim 2 and (4.59) Next we consider the first summand of the right-hand side of (4.58). As preparation, we note that for fixed g ∈ G, the curve γ Y : where the last equality follows by Lemma 4.6, Claim 1. Thus we obtain (4.60) Since the curve γ X : Ê ∋ t → exp(tX) ∈ G fulfills γ X (0) = e andγ(0) = X = X G/H (e),

Invariant Metric Covariant Derivatives
In this short subsection, we assume that G/H carries an invariant pseudo-Riemannian metric defined by an Ad(H)-invariant scalar product ·, · : m × m → Ê. We characterize all Ad(H)-invariant bilinear maps α : m × m → m such that ∇ α is an invariant metric covariant derivative with respect to the invariant pseudo-Riemannian metric corresponding to ·, · . To this end, we first recall that a covariant derivative ∇ on a manifold M is called compatible with the pseudo-Riemannian metric g ∈ Γ ∞ S 2 (T * M ) , or metric for short, if holds, see e.g. [12,Sec. 22.5].
Notation 4.21 In this subsection, we denote by g and g a pseudo-Riemannian metric on G and a fiber metric on Hor(G), respectively, while in the previous sections as well as in the sequel, we usually denote by g an element in a Lie group G. is skew-adjoint with respect to ·, · , i.e.
holds for all X, Y, Z ∈ m.
where holds by the definition of g ∈ Γ ∞ S 2 T * (G/H) . Since Z and Z are pr-related, we obtain by [10,Prop. 8.16] and (4.68) where we exploited that g A L i , A L j = A i , A j holds by the definition of g ∈ Γ ∞ S 2 Hor(G) * . Moreover, with ∇ Hor,α from Lemma 4.10, we compute where the second equality holds iff is satisfied for all i, j, k ∈ {1, . . . , N }. Since We now recall an expression for the Levi-Civita covariant derivative on a reductive homogeneous space G/H equipped with an invariant pseudo-Riemannian metric corresponding to the Ad(H)-invariant scalar product ·, · : m × m → Ê. This is the next proposition which is taken from [5,Sec. 23.6], where it is stated for the Riemannian case. However, since its proof only relies on the non-degeneracy of the invariant pseudo-Riemannian metric and its associated Ad(H)-invariant scalar product, it can be generalized to the pseudo-Riemannian setting. for all Z ∈ m.
Proposition 4.23 can be simplified for naturally reductive homogeneous spaces. This is the next corollary which can be seen as a reformulation of [5,Prop. 23.25] adapted to the pseudo-Riemannian setting.

Corollary 4.24
Let G/H be a naturally reductive homogeneous space. Then holds for all X, Y ∈ m.
Proof: This is proven in [5,Prop. 23.25]. Nevertheless, we include the proof here, as well. Since

Parallel Vector Fields along Curves
Having an expression for ∇ α on a reductive homogeneous space G/H in terms of horizontally lifted vector fields on G allows for determining the associated covariant derivative of vector fields along a given curve on G/H in terms of horizontal lifts, as well. In this subsection, an ODE for a specific curve in m is determined which is fulfilled iff the corresponding vector field along the given curve is parallel. Let be a curve and let Z : be a vector field along γ, i.e Z(t) ∈ T γ(t) (G/H), t ∈ I. denote a horizontal lift of γ with respect to the principal connection P ∈ Γ ∞ End(T G) from Proposition 3.12. It is well-known that g is unique up to the initial condition g(t 0 ) = g 0 ∈ G γ(0) . Furthermore, the curve g is defined on the whole interval I since principal connections are complete, see e.g. [12,Thm. 19.6]. Let Z : I → Hor(G) be the horizontal lift of Z along g, i.e.
Next we define the curves in m associated with g and Z, namely and We now consider the covariant derivative of Z along γ. This is next proposition which can be seen as a generalization of [7, Lem. 1], where we use the notation which has been introduced above.
) and Z(t) = z j (t)A L j (g(t)) (4.83) for some uniquely determined smooth functions x i , z j : I → Ê. Let α : m × m → m be an Ad(H)-invariant bilinear map and let ∇ α be the corresponding invariant covariant derivative on G/H. Then the associated covariant derivative of Z along γ lifted to a horizontal vector field along g : I → G is given by for all t ∈ I, where z : Proof: The proof is essentially given by applying Theorem 4.15. To this end, we define the vector field X : and we denote by X : I → T G the horizontal lift of X along g : I → G. Moreover, for fixed t 0 ∈ I, we extend X and Z to vector fields defined on an open neighbourhood O ⊆ G/H of γ(t 0 ). These vector fields are denoted by respectively. In particular, X(γ(t)) = X(t) =γ(t) and Z(t) = Z(γ(t)) is fulfilled for all t in a suitable open neighbourhood of t 0 in I. Moreover, their horizontal lifts X, Y ∈ Γ ∞ Hor(G) pr −1 (O) fulfill X(t) = X(g(t)) =ġ(t) and Z(t) = Z(g(t)).
These horizontal lifts can be expanded in the global frame A L 1 . . . , A L N of Hor(G). We write for t ∈ I in a suitable open neighbourhood of t 0 Similarly, we expand and where x i , z j : pr −1 (O) ⊆ G → Ê are uniquely determined smooth functions for i, j ∈ {1, . . . , N }. By construction x i (t) = x i (g(t)) and z j (t) = z j (g(t)) holds for all t in a suitable open neighbourhood of t 0 in I. We now use [11,Thm. 4.24] as well as Theorem 4.15 to compute the horizontal lift of the covariant derivative of Z along γ. We obtain for t ∈ I in a suitable neighbourhood of t 0 = ż(t) L (g(t)) + α(x(t), z(t)) L (g(t)).
Applying this argument for each t 0 ∈ I yields the desired result.
for t ∈ I.

Geodesics
In this short section, we consider geodesics on the reductive homogeneous space G/H with respect to an invariant covariant derivative ∇ α . Recall that a curve γ : I → G/H is a geodesic if the vector fieldγ : I → T (G/H) along γ is a parallel. Thus Corollary 4.27 can be used to obtain the following characterization of the geodesics on G/H with respect to ∇ α . We now apply Lemma 4.28 to a reductive homogeneous space equipped with the Levi-Civita covariant derivative defined by some invariant pseudo-Riemannian metric. Inspired by the well known characterization of geodesics on a Lie group equipped with a leftinvariant metric given in [1,Ap. B], see also [4,Sec. 4] for a discussion in the complex setting, we obtain the next corollary which generalizes the description of geodesics on Lie groups equipped with left-invariant metrics.
Corollary 4.29 Let G/H be a reductive homogeneous space and let ·, · : m × m → Ê be an Ad(H)-invariant scalar product. Moreover, let ∇ LC denote the Levi-Civita covariant derivative defined by the invariant metric on G/H corresponding to ·, · . Let γ : I → G/H be a curve in G/H and g : I → G be a horizontal lift of γ. Define x : I ∋ t → x(t) = (T e ℓ g(t) ) −1ġ (t) ∈ m. Then γ : I → G/H is a geodesic with respect to ∇ LC iff the ODĖ x(t) = (pr m • ad x(t) ) * (x(t)) (4.87) is satisfied for all t ∈ I. Here (pr m • ad X ) * : m → m denotes the adjoint with respect to ·, · of the linear map defined for fixed X ∈ m by pr m • ad X : m → m. for all X, Y, Z ∈ m, where (pr m • ad X ) * and (pr m • ad Y ) * denote the adjoints of the linear maps (pr m • ad Y ) and (pr m • ad Y ) with respect to ·, · , respectively. Since ·, · is nondegenerated, we can rewrite (4.89) equivalently as U (X, Y ) = − 1 2 (pr m • ad X ) * (Y ) + (pr m • ad Y ) * (X) . Thus we obtain α(X, X) = − 1 2 [X, X] m + U (X, X) = −(pr m • ad X ) * (X), for all X ∈ m. Now Lemma 4.28 yields the desired result.
As indicated above, by applying Corollary 4.29 to a Lie group equipped with a left-invariant pseudo-Riemannian metric considered as the reductive homogeneous space G ∼ = G/{e}, one obtains the following corollary concerning geodesics on G. Its statement is well-known and can be found in [1,Ap. 2]. We also refer to [4,Sec. 4] for a discussion of this characterization of geodesics in the complex setting, where it is named Euler-Arnold Formalism.
Corollary 4.30 Let G be a Lie group equipped with a left-invariant metric defined by the scalar product ·, · : g × g → Ê. Then g : I → G is a geodesic iff the curve x : I ∋ t → x(t) = (T e ℓ g(t) ) −1ġ (t) ∈ g satisfiesẋ (t) = (ad x(t) ) * (x(t)) (4.90) for all t ∈ I. Here (ad X ) * : g → g denotes the adjoint of ad X : g → g with respect to ·, · , where X ∈ g is fixed.
Proof: Clearly, the Lie group G equipped with the left-invariant metric defined by the scalar product ·, · : g×g → Ê can be viewed as the reductive homogeneous space G/H for H = {e} with reductive decomposition g = {0} ⊕ g equipped with the pseudo-Riemannian metric defined by the Ad({e})-invariant scalar product ·, · on g. Thus the assertion follows by Corollary 4.29 due to pr m = id g .

Canonical Invariant Covariant Derivatives
We now relate two particular invariant covariant derivatives on G/H to the canonical affine connections of first and second kind from [13,Sec. 10]. To this end, we list the two properties concerning invariant covariant derivatives which correspond to the properties of invariant affine connections from [13, Sec. 10, (A1) and (A2)]. This is the next definition. 1. The curves γ X : Ê ∋ t → pr(exp(tX)) ∈ G/H are geodesics with respect to ∇ α for all X ∈ m.
2. The curves γ X : Ê ∋ t → pr(exp(tX)) ∈ G/H are geodesics with respect to ∇ α for all X ∈ m and the parallel transport of T e pr Z ∈ T pr(e) (G/H) along γ X with respect to ∇ α is given by Z : Ê ∋ t → T exp(tX) pr •T e ℓ exp(tX) Z ∈ T (G/H) for all Z ∈ m.
Proof: Let X ∈ m be arbitrary. We define the curve γ X : Ê ∋ t → pr(exp(tX)) ∈ G/H. Obviously, the curve Ê ∋ t → exp(tX) ∈ G is a horizontal lift of γ. Define x : Ê → m by x(t) = (T e ℓ exp(tX) ) −1 • (T e pr m ) −1γ (t). Clearly, x(t) = X holds for all t ∈ Ê. By Lemma 4.28, the curve γ : I → G/H is a geodesic with respect to ∇ α iff α(X, X) = 0 holds, i.e. Claim 1 is shown. It remains to prove Claim 2. To this end, let Z ∈ m be arbitrary. We now define the vector field Z : Ê ∋ t → T exp(tX) pr •T e ℓ exp(tX) Z ∈ T (G/H) along the curve γ X : Ê ∋ t → pr(exp(tX)) ∈ G/H. Next we consider the curve z : Ê → m given by  The corresponding invariant covariant derivative ∇ α is the unique invariant covariant derivative on G/H which is torsion free and satisfies Definition 4.31, Claim 1.