Rolling Reductive Homogeneous Spaces

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space $G / H$ with reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$, we consider rollings of $\mathfrak{m}$ over $G / H$ without slip and without twist, where $G / H$ is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space $Q$ which is tangent to a certain distribution. By considering a $H$-principal fiber bundle $\overline{\pi} \colon \overline{Q} \to Q$ over the configuration space equipped with a suitable principal connection, rollings of $\mathfrak{m}$ over $G / H$ can be expressed in terms of horizontally lifted curves on $\overline{Q}$. The total space of $\overline{\pi} \colon \overline{Q} \to Q$ is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of $\mathfrak{m}$ over $G / H$ as solutions of an explicit, time-variant ordinary differential equation (ODE) on $\overline{Q}$, the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in $G / H$ is the projection of a one-parameter subgroup in $G$. Lie groups and Stiefel manifolds are discussed as examples.


Introduction
Meanwhile, there is a vast literature on rolling manifolds without slip and without twist. First, we mention some works, where concrete expressions for extrinsic rollings of certain submanifolds of (pseudo-)Euclidean vector spaces over their affine tangent spaces are derived. Using the definition from [28, Ap. B] as starting point, extrinsic rollings of spheres S n ⊆ Ê n+1 , real Grassmann manifolds Gr n,k ⊆ Ê n×n sym and special orthogonal groups SO(n) ⊆ Ê n×n over their affine tangent spaces are studied in [12]. In a similar context, the Stiefel manifold St n,k ⊆ Ê n×k , endowed with the Euclidean metric, is investigated in [9] while rollings of pseudo-orthogonal groups are considered in [3]. For these works, the need to solve interpolation problems on these submanifolds in various applications seems to serve as a motivation. Indeed, the rolling and unwrapping technique from [12], see also of given points on the manifolds S n , Gr n,k and SO(n), where the velocities at the initial and final point are prescribed. This algorithm relies on having an explicit expression for the rolling of the manifold over its affine tangent space along a curve joining the initial point with the final point.
Beside these works, there is the paper [6], where a notion of intrinsic rolling of an oriented Riemannian manifold M over another oriented Riemannian manifold M is introduced assuming dim(M ) = dim( M ). In [21], this notion of intrinsic rolling is generalized to pseudo-Riemannian manifolds. A further generalization can be found in [7,Sec. 7] and [17, p. 35], where the Levi-Civita covariant derivatives coming from the pseudo-Riemannian metrics on M and M are replaced by arbitrary covariant derivatives on M and M , respectively.
In this text, we investigate the following situation. Let G be a Lie group and H ⊆ G a closed subgroup such that G/H is a reductive homogeneous space with a fixed reductive decomposition g = h⊕m. Then G/H can be equipped with an invariant covariant derivative corresponding to an invariant affine connection from [24]. Motivated by the study of rollings of (pseudo-Riemannian) symmetric spaces over flat spaces in [15], we consider rollings of m over G/H. Here we generalize the above mentioned definition proposed in [7] and [17, p. 35] slightly in order take additional structures of the involved manifolds into account. In particular, this definition allows for considering rollings of not necessarily oriented manifolds.
Moreover, if one is interested in getting rather simple formulas describing the rollings, it might be convenient to consider rollings of m over G/H with respect to the canonical covariant derivative of first or second kind on G/H. These covariant derivatives can be defined independently of a pseudo-Riemannian metric although they are in some sense similar the Levi-Civita covariant derivatives on naturally reductive homogeneous spaces or pseudo-Riemannian symmetric spaces, respectively.
We now give an overview of this text. In Section 2, we start with introducing some notations and recalling some definitions and well-known facts related to Lie groups and principal fiber bundles. Moreover, we recall some facts on reductive homogeneous spaces with an emphasize on invariant covariant derivatives.
In Section 3, we briefly recall the notion of rolling intrinsically a manifold M over another manifold M of equal dimension from the literature. More precisely, as already announced above, a slightly generalized definition of intrinsic rolling is introduced.
As preparation to determine the configuration space for the intrinsic rollings considered in Section 5, an explicit description of the frame bundle of a reductive homogeneous space G/H is needed. Therefore frame bundles of reductive homogeneous spaces are investigated in Section 4. Here we first consider a more general situation. The frame bundle of a vector bundle associated to a H-principal fiber bundle P → M is identified with an other fiber bundle associated to P → M . Afterwards, reductive homogeneous spaces are treated as a special case.
In Section 5, we turn our attention to rollings of a reductive homogeneous space G/H with reductive decomposition g = h ⊕ m. We consider the intrinsic rolling of m over G/H with respect to an invariant covariant derivative ∇ α . To this end, the configuration space Q → m × G/H is investigated in detail. Here we determine a H-principal fiber bundle π : Q → Q over Q which is equipped with a suitable principal connection. Its total space is given by Q = m × G × G(m), where G(m) ⊆ GL(m) is a closed subgroup, i.e. the manifold Q is a product of Lie groups.
For a fixed invariant covariant derivative ∇ α on G/H defined by an Ad(H)-invariant billinear map α : m × m → m, we determine a distribution D α on Q that projects to a distribution D α on Q with the following property. A curve q : I → Q is horizontal with respect to D α iff it is a rolling of m over G/H with respect to ∇ α . Moreover, horizontal lifts of curves on Q with respect to the principal connection on π : Q → Q mentioned above are horizontal with respect to D α iff they are horizontal with respect to D α . In particular, this fact allows for characterizing rollings of m over G/H in terms of an ODE on Q. More precisely, for a prescribed control curve u : I → m, we obtain an explicit, time-variant ODE on Q = m × G × G(m) whose solutions projected to Q are rollings of m over G/H with respect to ∇ α . This ODE can be seen as a generalization of the kinematic equation for rollings of oriented pseudo-Riemannian symmetric spaces over flat spaces from [15,Sec. 4.2].
In Subsection 5.4, we turn our attention to rollings of m over G/H with respect to the canonical covariant derivative of first and second kind such that the development curve is of the form I ∋ t → pr(exp(tξ)) ∈ G/H with some ξ ∈ g, i.e. a projection of a not necessarily horizontal one-parameter subgroup in G. For this special case, an explicit solution of the kinematic equation is obtained.
We end this text by discussing intrinsic rollings of Lie groups and Stiefel manifolds as examples.

Notations, Terminology and Background
In this section, we introduce the notation and terminology that is used throughout this text. Moreover, some facts concerning Lie groups and principal fiber bundles are recalled. We end this section by discussing reductive homogeneous spaces with an emphasize on invariant covariant derivatives.

Notations and Terminology
We start with introducing some notations and terminology concerning differential geometry. This subsection is an extended version of [27,Sec. 2] mostly copied word by word.
Notation 2.1 Throughout this text we follow the convention in [25,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, L X T denotes the Lie derivative. If x : N → Ê is a smooth function, we write f * x = x • f : M → Ê for its pull-back by f : M → N . 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. Next assume that f : M → N is a local diffeomorphism. Then the pull-back of the tensor field T ∈ Γ ∞ (T * N ) ⊗k ⊗ (T N ) ⊗ℓ by f is denoted by f * T , as well.
We now consider a fiber bundle pr : P → M over M . Its vertical bundle is denoted by Ver(P ) = ker(T pr) ⊆ T P . We write Hor(P ) ⊆ T P for a horizontal bundle, i.e. a subbundle of T P fulfilling Ver(P ) ⊕ Hor(P ) = T P . If pr P : P → M and pr Q : Q → M are fiber bundles over the same manifold M with typical fiber F P and F Q , respectively, their fiber product is denoted by P ⊕ Q → M . It is the fiber bundle over M given by P ⊕ Q = {(p, q) ∈ P × Q | pr P (p) = pr Q (q)} (2.1) with typical fiber F P × F Q . Next let S 1 × · · · × S k be a product of sets and let i ∈ {1, . . . , k}. Then we denote by the projection onto the i-th factor. We now recall a well-known fact on surjective submersions. This is the next lemma, see e.g. [19,Thm. 4.29], which is used frequently without referencing it explicitly. Concerning the regularity of curves on manifolds, we use the following convention.

Lie groups
Copying and extending [27,Sec. 3.1], we now introduce 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. The left translation by an element g ∈ G is denoted by ℓ g : G → G, h → ℓ g (h) = gh (2.4) and we write r g : G → G, h → r g (h) = hg (2.5) for the right translation by g ∈ G. The conjugation by an element g ∈ G is given by Conj g : G → G, h → Conj g (h) = (ℓ g • r g −1 )(h) = (r g −1 • ℓ g )(h) = ghg −1 (2.6) and the adjoint representation of G is defined as Ad : G → GL(g), g → Ad g = ξ → Ad g (ξ) = T e Conj g ξ . (2.7) Moreover, we denote the adjoint representation of g by ad : g → gl(g), ξ → η → ad ξ (η) = [ξ, η] . (2.8) Next we recall [4,Def. 19.7]. A vector field X ∈ Γ ∞ (T G) is called left-invariant or right-invariant if for all g, k ∈ G T k ℓ g X(k) = X(ℓ g (k)) or T k r g X(k) = X(r g (k)), (2.9) 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. (2.10) The exponential map of the Lie group G is denoted by exp : g → G. (2.11) One has for ξ ∈ g and t ∈ Ê d dt exp(tξ) = T e ℓ exp(tξ) ξ = T e r exp(tξ) ξ (2.12) by the proof of [4,Prop. 19.5].
Next we recall that the tangent map of the group multiplication m : [22,Lem. 4.2]. The tangent map of the inversion inv : G ∋ g → inv(g) = g −1 ∈ G reads for all g ∈ G and v g ∈ T g G, see e.g. [22,Cor. 4.3]. We now introduce the notation for some Lie groups that play a crucial role in this text.
Notation 2.5 Let V be a finite dimensional Ê-vector space. We write GL(V ) for the general linear group of V . If V is a pseudo-Euclidean vector space, i.e. V is endowed with a scalar product ·, · : V × V → Ê, we denote the corresponding pseudo-orthogonal group by O(V, ·, · ). Moreover, we often write O(V ) = O(V, ·, · ) for short. Similarly, the special (pseudo-)orthogonal group is denoted by SO(V ) = SO(V, ·, · ). More generally, a closed subgroup of GL(V ), which is not further specified, is often denoted by G(V ) and we write g(V ) ⊆ gl(V ) for the corresponding Lie algebra. Sometimes, the exponential map of G(V ) is denoted by In the sequel, it is often convenient to denote the evaluation of A ∈ GL(V ) at v ∈ V by Av instead of writing A(v).

Principal Fiber Bundles
Next we recall some well-known facts on principal fiber bundles and introduce some notations. For general facts on principal fiber bundles we refer to [22,  and we denote for fixed h ∈ H by (· ⊲ h) : P ∋ p → p ⊳ h ∈ P the induced diffeomorphism.
Next, let η ∈ h. Then η P ∈ Γ ∞ (T P ) denotes the fundamental vector field associated to the principal action. For p ∈ P , it is given by The vertical bundle Ver(P ) = ker(T pr) ⊆ T P of P → M is fiber-wise given by is an isomorphism of vector bundles covering id P , see e.g. [26,Lem. 1.3.1] or [22,Sec. 18.18].
Recall that a complement of Ver(P ), i.e. a subbundle Hor(P ) ⊆ T P fulfilling Hor(P ) ⊕ Ver(P ) = T P is called horizontal bundle. It is well-known that such a complement defines a unique connection on P , i.e. an endomorphism P ∈ Γ ∞ End(T P ) such that P 2 = P and im(P) = Ver(P ) as well as im(P) = Hor(P ) holds. This fact can be regarded as a consequence of [22,Sec. 17.3]. Moreover, such a connection on the principal fiber bundle P → M is in one-to-one correspondence with a h-valued one-form ω ∈ Γ ∞ (T * P ) ⊗ h via holds for all h ∈ H or equivalently · ⊳h * P = P is satisfied for all h ∈ H, see e.g. [22,Sec. 19.1]. We now recall from [22,Sec. 19.1] how a principal connection is related to the corresponding connection one-form given by (2.20). This is the next lemma.
2. For each h ∈ H one has (· ⊳ h) * ω p (v p ) = Ad h −1 ω p (v p ) for all p ∈ P and v p ∈ T p P .
Conversely, a h-valued one-form ω ∈ Γ ∞ (T * P ) ⊗ h fulfilling Claim 1 and Claim 2 defines a principal connection on P → M via for p ∈ P and v p ∈ T p P with the map (p ⊳ ·) : H ∋ h → p ⊳ h ∈ P for fixed p ∈ P .
Next we recall the notion of reductions of principal fiber bundles, see e.g. [22,Sec. 18.6]. Let P → M be a H-principal fiber bundle. Then a H 2 -principal fiber bundle P 2 → M is called a reduction of P if there is a morphism of Lie groups f : H 2 → H and a morphism Ψ : P 2 → P of principal fiber bundles along f covering id M : M → M . In particular, holds for all h 2 ∈ H 2 and p 2 ∈ P 2 . In the sequel, only reductions of H-principal fiber bundles along the canonical inclusion of a closed subgroup H 2 ⊆ H into a Lie group H will play a role. Furthermore, we need the notion of an associated bundle which we recall briefly from [22,Sec. 18.7]. Let F be some manifold and let ⊲ : H × F → F be a smooth action of H on F from the left. Then the corresponding associated bundle is denoted by (2.24) whose elements are given by Here [p, s] denotes the equivalence class of (p, s) ∈ P × F defined by the H-action on P × F given by The projection π : P × H F → M , sometimes denoted by π P × H F : P × H F → M to refer to P × H F explicitly, is given by π([p, s]) = pr P (p), where pr P : P → M denotes the projection of the principal fiber bundle. Moreover, we often write for the H-principal fiber bundle over the associated bundle P × H F , where the principal action is given by (2.26) and the map π in (2.27) is the canonical projection, i.e. π(p, f ) = [p, f ] for (p, f ) ∈ P × F . We also denote this projection by π P ×F : P × F → P × H F to refer to P × F explicitly. Moreover, we will use the following identification of the tangent bundle of an associated [22,Sec. 18.18]. Here T P is considered as T H-principal fiber bundle over T M with principal action T ⊳ : [22,Sec. 18.18], and T H acts on T F via the tangent map of the H-action on F denoted by T ⊲ : . Finally, we introduce some notations concerning frame bundles of vector bundles. We refer to [22,Sec. 18.11] for general information on frame bundles.

Notation 2.8
The frame bundle of a vector bundle E → M with typical fiber V is denoted by GL(V, E) → M . If E is equipped with a not necesarrily positive definite fiber metric, we denote the corresponding (pseudo-)orthogonal frame bundle by O(V, E) → M . More generally, let G(V ) ⊆ GL(V ) be a closed subgroup of the general linear group GL(V ). Then a G(V )-reduction of GL(V, E) along the canonical inclusion G(V ) → GL(V ) is often denoted by G(V, E) if it exists. We write pr G(V,E) : G(V, E) → M for the bundle projection.

Reductive Homogeneous Spaces
In this subsection, we recall some well-known facts on reductive homogeneous spaces and introduce the notation that is used throughout this text. Afterwards, invariant covariant derivatives are considered.

Some General Facts
We now recall some general facts concerning reductive homogeneous spaces mostly by copying [27,Sec. 3.2] word by word. We refer to [4,Sec. 23.4] or [25,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 G-action on G/H from the left, where g · H ∈ G/H denotes the coset defined by g ∈ G. Borrowing the notation from [4, p. 676], for fixed g ∈ G, the associated diffeomorphism is denote by In addition, we write pr : G → G/H, g → pr(g) = g · H (2.31) for the canonical projection. Since reductive homogeneous spaces play a central role in this text, we recall their definition from [4,Def. 23.8], see also [24,Sec. 7]   is fulfilled for all h ∈ H.
Notation 2.11 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).
holds for all g ∈ G. In the next lemma which is taken from [25,Chap. 11,Prop. 22], see also [4,Prop. 23.22]  holds for all X, Y, Z ∈ m.
The following lemma can be considered as a generalization of [4,Prop. 23.29 (1)-(2)] to pseudo-Riemannian metrics and Lie groups which are not necessarily connected.
Lemma 2.14 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 [4,Prop. 23.29 (1)-(2)] by taking the assumption h ⊕ h ⊥ = h ⊕ m = g into account.
Remark 2.15 Inspired by the terminology in [4, Sec. 23.6, p. 710], we refer to the naturally reductive spaces from Lemma 2.14 as normal naturally reductive spaces.
We now consider another special class of reductive homogeneous spaces. To this end, we state the following definition which can be found in [8, p. 209].
Definition 2.16 Let G be a connected Lie group and let H be a closed subgroup. Then (G, H) is called a symmetric pair if there exists a smooth involutive automorphism σ : G → G, i.e. an automorphism of Lie groups fulfilling σ 2 = σ, such that (H σ ) 0 ⊆ H ⊆ H σ holds.
Here H σ denotes the set of fixed points of σ and (H σ ) 0 denotes the connected component of H σ containing the identity e ∈ G.
Inspired by the terminology used in [4,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. [24,Sec. 14]. Note that the definition in [24, Sec. 14] does not require an invariant pseudo-Riemannian metric on G/H. The next lemma, see e.g. [24,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 [4,Prop. 23.33] for a proof.

Reductive Decompositions and Principal Connections
We now consider the principal connection defined on the principal fiber bundle pr : G → G/H by means of the reductive decomposition This subsection can be seen as a short version of [27,Sec. 3.3] mostly copied word by word. Let G be a Lie group and H ⊆ G a closed subgroup. It is well-known that pr : G → G/H carries the structure of a H-principle fiber bundle, see e.g. [22,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. It is well-known that this reductive decomposition can be used to obtain a principal connection on pr : G → G/H, see [16,Thm. 11.1]. Before we consider this principal connection on the principal fiber bundle G → G/H, we comment on its vertical bundle. For fixed g ∈ G the vertical bundle of pr : G → G/H is given by according to (2.18). The next proposition describes the principal connection on pr : G → G/H defined by the reductive decomposition g = h ⊕ m mentioned above.
Proposition 2.20 Consider pr : G → G/H as a H-principal fiber bundle, where G/H is a reductive homogeneous space with reductive decomposition g = h ⊕ m and define Hor(G) ⊆ T G fiber-wise by Hor(G) g = (T e ℓ g )m, g ∈ G. (2.44) 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.

Invariant Covariant Derivatives
In this subsection which is mostly based on [27, Sec. 4.1], partially copied word by word, we recall some facts on invariant covariant derivatives on reductive homogeneous spaces. We point out that invariant covariant derivatives correspond to the well-known invariant affine connections from [24]. In the sequel, let G/H be a reductive homogeneous space with a fixed reductive decomposition g = h ⊕ m. We start with defining Ad(H)-invariant bilinear maps and invariant covariant derivatives.
holds for all X, Y ∈ m and h ∈ H.
holds for all g ∈ G and X, Y ∈ Γ ∞ T (G/H) , where (τ g ) * X denotes the push-forward of X by τ g : In order to establish the one-to-one correspondence between Ad(H)-invariant bilinear maps and invariant covariant derivatives, we introduce the following notation. Let X ∈ g. We denote by X G/H ∈ Γ ∞ T (G/H) the fundamental vector field associated with the action τ : G × G/H → G/H, i.e. X G/H is defined by  ] that corresponds to α is related to ∇ α by ∇ α X Y = t α (Y )(X) for all X, Y ∈ Γ ∞ T (G/H) according to [27,Prop. 4.19].
According to [27,Sec. 4.1.2] the invariant covariant derivative ∇ α from Definition 2.24 can be expressed in terms of horizontally lifted vector fields as follows. Let X, Y ∈ Γ ∞ T (G/H) and let X, Y ∈ Γ ∞ Hor(G) denote their horizontal lifts with respect to the principal connection from Proposition 2.20. Moreover, let {A 1 , . . . , A N } ⊆ m be a basis and let A L 1 , . . . , A L N ∈ Γ ∞ Hor(G) be the associated left invariant frame. Writing X = x i A L i and Y = y j A L j with uniquely determined smooth functions x i , y j : G → Ê for i, j ∈ {1, . . . , N } one obtains by [27,Thm. 4.16] for the horizontal lift ∇ α where Einstein summation convention is used, as usual. This expression paves the way for characterizing parallel vector fields along curves on G/H with respect to ∇ α . Indeed, we have the next proposition which is a reformulation of [27,Cor. 4.27].
is satisfied for all t ∈ I.

Metric Invariant Covariant Derivatives
If G/H is equipped with an invariant pseudo-Riemmannian metric, one has the following characterization of invariant metric covariant derivatives which is copied from [27,Prop. 4.22].
holds for all X, Y, Z ∈ m.
Moreover, we state the following reformulation of [27,Re. 4.25] concerning the Levi-Civita covariant derivative of a naturally reductive homogeneous space.

Canonical Invariant Covariant Derivatives
In this subsection, we consider the so-called canonical invariant covariant derivatives of first and second kind which correspond to the canonical affine connections from [24,Sec. 10]. Here we copy and adapt some parts of [27,Sec. 4.6].

Definition 2.29
The canonical covariant derivative of first kind ∇ can1 is defined by the Ad(H)-invariant bilinear map given by The canonical covariant derivative of second kind ∇ can2 corresponds to the Ad(H)-invariant bilinear map These canonical covariant derivatives correspond to the Levi-Civita covariant derivatives on certain pseudo-Riemannian homogeneous spaces. More precisely, we have the next remarks which are copied from [27].

Intrinsic Rolling
In this section, a notion of rolling intrinsically a manifold M over another manifold M of equal dimension dim(M ) = n = dim( M ) is recalled from the literature and slightly generalized. As preparation to define the configuration space, we state the following lemma which can be regarded as a slight generalization of the definition of the configuration space in [6,Sec. 3.1]. In particular, the definition of the map Ψ in Lemma 3.1, Claim 2, below, is very similar to [6, Eq. (4)].
be defined as the quotient of G(V, E) × G(V, E) by the diagonal action of G(V ), where the action on each component is given by the G(V )-principal action. Moreover, define Then the following assertions are fulfilled: is bijective.
Proof: The action , is free and proper since the action on each component is free and proper. Thus respectively. Locally, one obtains for the principal action for A ∈ G(V ) [22,Sec. 18,p. 211]. We now define the local trivialization φ : The map φ is well-defined. Indeed, we obtain by (3.5) for Moreover, the canonical projection pr : is a surjective submersion and the map is smooth as the composition of smooth maps. Obviously, φ = pr • φ holds. Hence φ is smooth, too. Moreover, φ is bijective. Indeed, the map yields the inverse of φ since one verifies by a straightforward computation Next we define the map which is smooth as the composition of smooth maps. Then φ −1 = pr•φ −1 is clearly fulfilled. Hence φ −1 is smooth by the smoothness of φ −1 since pr is a surjective submersion. Thus φ is a diffeomorphism. This yield the desired result since for every point (x, x) ∈ M × M , we can construct a local trivialization φ : . By the fiber-wise transitivity of the principal G(V )-actions on G(V, E) and G(V, E), respectively, we obtain for A, B ∈ G(V ) i.e. Ψ is surjective. This yields the desired result.
After this preparation, we consider intrinsic rollings. Let M and M be two manifolds with dim(M ) = n = dim( M ). Moreover, let G(Ê n ) ⊆ GL(Ê n ) be a closed subgroup and assume that the frame bundles GL(Ê n , T M ) → M and GL(Ê n , T M ) → M admit both a G(Ê n )-reduction along the canonical inclusion G(Ê n ) → GL(Ê n ). These reductions are denoted by respectively. In this section, we denote by the G(Ê n )-fiber bundle over M × M obtained by applying Lemma 3.1 to the frame bundles from (3.6). We now define a notion of rolling of M over M intrinsically, where M and M are both equipped with a covariant derivative ∇ and ∇, respectively.
with projection (x, x) = π • q : I → M × M such that the following conditions are fulfilled: 1. No slip condition:˙ x(t) = q(t)ẋ(t) for all t ∈ I.

No twist condition:
Here Lemma 3.1, Claim 2 is used to identify q(t) with the linear isomorphism q(t) : , as well. We call the curve x : I → M rolling curve. The curve x : I → M is called development curve. The curve q : I → Q is often called rolling for short.
The next remark yields an other perspective on the intrinsic rollings from Definition 3.2.   Studying properties of rollings in the sense of Definition 3.2 for general manifolds is out of the scope of this text. However, in Section 5 below, we discuss intrinsic rollings in the context of reductive homogeneous spaces in detail.

Frame Bundles of Associated Vector Bundles
In this section, we identify (certain reductions of) the frame bundle of a reductive homogeneous space G/H with certain principal fiber bundles obtained as associated bundles of the H-principal fiber bundle pr : G → G/H. We point out that the results of this section might be well-known since the statement of Corollary 4.11 can be found as an exercise in the German book [2, Ex. 2.7]. However, we were not able to find a reference including a proof. Hence we provide one in this section in order to keep this text as self-contained as possible. Here we first start with a more general situation that is applied to reductive homogeneous spaces later. We first determine (certain reductions of) the frame bundles of vector bundles given as associated bundles of some principal fiber bundle.

Frame Bundles of Associated Vector Bundles
Let P → M be a H-principal fiber bundle. We describe (reductions of) the frame bundle of a vector bundle associated to P in terms of another fiber bundle associated to P . To this end, we state the following lemma as preparation.
Lemma 4.1 Let P → M be a H-principal fiber bundle and let ρ : Then the following assertions are fulfilled: smoothly from the left.

The map
denoted by the same symbol as the principal action ⊳ : P × H → P , yields a welldefined, smooth, free and proper G(V )-right action on the associated bundle into a G(V )-principal fiber bundle, where π denotes the canonical projection.

The map
is a diffeomorphism such that φ • π = π holds. Moreover, is an isomorphism of G(V )-principal fiber bundles covering φ.
for all p ∈ P , h ∈ H and A, B ∈ G(V ). Next we show that ⊳ is smooth. To this end, we consider the diagram where π : P × G(V ) → (P × G(V ))/H = P × H G(V ) denotes the canonical projection and ⊳ is given by which is clearly a smooth and free G(V )-right action on P × G(V ). Moreover, the action ⊳ is proper since the G(V )-action on G(V ) by right translations is proper, see e.g. [5,Prop. 9.29].
The map π × id G(V ) is a surjective submersion and (π × id G(V ) ) • ⊳ is smooth as the composition of smooth maps. Thus the action ⊳ is smooth since the diagram (4.7) commutes. Next holds proving that ⊳ is free. We now show that ⊳ is proper. To this end, we use the characterization of a proper Lie group action in terms of sequences, see e.g. [22,Sec. 6.20]. Let by the continuity of the local section s : U → P × G(V ) and the convergence of Moreover, let (B i ) i∈AE be a sequence in G(V ) such that the sequence defined by Clearly, holds by (4.8). Next we choose a local sections s 2 : We define the sequence which is a smooth map as the composition of smooth maps. The definition of ( p i , C i ) i∈AE in (4.9) implies which converges by the continuity of Θ as well as the convergence of the sequences where we used (4.10) to obtain last equality. Since the action ⊳ : showing that φ is well-defined. We now consider the diagrams and where φ −1 is given by In addition, φ and φ −1 are smooth since (4.12) and (4.13) commute and π as well as π are both surjective submersions. Hence the commutativity of (4.12) implies that id P × H G(V ) is indeed an isomorphism of G(V )-principal fiber bundles over φ as desired.
as an G(V )-principal fiber bundle over M which is denoted by the same symbol as the associated bundle, i.e. from now on, we write which clearly commutes. Since the map π P ×G(V ) is a surjective submersion and π GL(V ) • ι P ×G(V ) is smooth as the composition of smooth maps, the map ι P × H G(V ) is smooth, as well, because (4.16) commutes. Clearly, the map showing that ι P × H G(V ) is a morphism of principal fiber bundles along the canonical inclu- The next proposition shows that π : P × H GL(V ) → M can be identified with the frame bundle of the associated vector bundle P × H V → M , where H acts on V via the representation viewed as the left action as an open subset of the morphism bundle is clearly linear. In addition, this map is invertible and its inverse is given by as well as Next we show that Ψ is a morphism of principal fiber bundles over id M . Clearly, We now show that Ψ is smooth. To this end, let P × H End(V ) → M denote the vector bundle associated to the H-principal fiber bundle pr : from the left. We now define the map An argument analogously to the one at the beginning of this proof, showing that Ψ is welldefined, proves that the map Ψ is well-defined, i.e. Ψ is independent of the representative Next we show that Ψ is a smooth morphism of vector bundles. To this end, we prove that Then the desired properties of Ψ follow by [19,Lem. 10.29].
is fulfilled for all x ∈ U . By this notation, we obtain for ( Hence the map Ψ(s) U ×V is smooth as the composition of smooth maps by (4.21). Thus Ψ(s) is smooth since x 0 ∈ M is arbitrary.
Next we prove the C ∞ (M )-linearity of Ψ. Let s 1 , s 2 ∈ Γ ∞ P × H End(V ) be two sections point-wise given by Here we assume without loss of generality that their first component is represented by the same element p(x) ∈ P for all x ∈ M . Moreover, let f, g ∈ C ∞ (M showing the C ∞ (M )-linearity of Ψ by its definition in (4.20). Hence Ψ is indeed a smooth morphism of vector bundles by [19,Lem. 10.29].
In order to prove the smoothness of Ψ, we consider the map whose smoothness can be proven analogously to the proof of Corollary 4.3 by exploiting the smoothness of the canonical inclusion P × GL(V ) → P × End(V ). We now obtain for Thus Ψ = Ψ • i is smooth as the composition of smooth maps. It remains to show that Ψ is an isomorphism of GL(V )-principal fiber bundles. To this end, we recall that the GL(V )-action on GL(V, P × H V ) is given by composition from the right, see e.g. [22,Sec. 18.11]. Thus we have for [p, A] ∈ P × H GL(V ) and B ∈ GL(V ) as well as proving that Ψ is a morphism of GL(V )-principal fiber bundles over id M : M → M . Therefore it is an isomorphism of principal fiber bundles by [5,Prop. 9.23].
Assuming that P × H GL(V ) admits a reduction as in Corollary 4.3, we obtain a reduction of GL(V, P × H V ).
holds for all v ∈ V by the definition of Ψ. Hence χ is an isomorphism of GL(V )-principal fiber bundles which covers the diffeomorphism φ : M → N .

Principal Fiber Bundles over Frame Bundles and Principal Connections
Since the G(V )-principal fiber bundle π : P × H G(V ) → M is obtained as a fiber bundle associated to the H-principal fiber bundle P → M , we have the H-principal fiber bundle π : P × G(V ) → P × H G(V ) over P × H G(V ). Given a principal connection on P → M , we construct a principal connection on π : P × G(V ) → P × H G(V ). This construction will be applied to the configuration space of an intrinsic rolling of a reductive homogeneous space in Proposition 5.4 below.
denote the induced morphism of Lie algebras. Consider the H-principal fiber bundle over the associated bundle π : Moreover, let P ∈ Γ ∞ End(T P ) be a principal connection on pr : P → M with corresponding connection one-form ω ∈ Γ ∞ (T * M ) ⊗ h. Then the following assertions are fulfilled: 1. The vertical bundle Ver(P × G(V )) ⊆ T (P × G(V )) ∼ = T P × T G(V ) is fiber-wise given by where (p, A) ∈ P × G(V ).

Defining
yields a principal connection on π : 3. Let q : I ∋ t → q(t) = (p(t), A(t)) ∈ P × G(V ) be a curve which is horizontal with respect to the principal connection P. Then the curve p : I → P given by the first component of q is horizontal with respect to the principal connection P on P → M .
Proof: First we recall that ρ ′ : h → g(V ) is indeed a morphism of Lie algebras, see e.g. [22,Lem. 4.13]. Next we prove Claim 1. To this end, we compute for (p, A) ∈ P ×G(V ) We now prove Claim 2. Obviously, P ∈ Γ ∞ End(T (P × G(V ))) holds. Next we show that P is a projection, i.e. P 2 = P is fulfilled. By using the correspondence of P and ω from (2.20) as well as P 2 = P, we calculate for p ∈ P and v p ∈ T p P (4.30) Using (4.30) and P 2 = P, we have for (p, Moreover, im(P) = Ver(P × G(V )) holds by im(P) = Ver(P ) and the characterization of the vertical bundle in (4.27).
We now show that P corresponds to ω. To this end, let η ∈ h and denote by η P ×G(V ) ∈ Γ ∞ T (P ×G(V )) the corresponding fundamental vector field associated to the H-principal action given by , (p, A) ∈ P × G(V ). By this notation and the definition of ω in (4.29), we obtain (4.31) Moreover, denoting by η P ∈ Γ ∞ (T P ) the fundamental vector field on P defined by η ∈ h, as usual, we compute since ω, being the connection one-form associated to P, fulfills ω(η P ) = η for all η ∈ h. Thus ω is the connection one-form corresponding to the connection P due to (4.31) and (4.32). In order to show that P is a principal connection, we show that ω has the desired equivarienceproperty. By exploiting that ω ∈ Γ ∞ (T * G) ⊗ h is a principal connection one-form, we compute for h ∈ H It remains to show Claim 3. Let q : I ∋ t → q(t) = (p(t), A(t)) ∈ P ×G(V ) be horizontal with respect to P. Then holds. In particular, this implies P p(t) (ṗ(t)) = 0. Hence p : I → P is horizontal with respect to the principal connection P on P → M .

Frame Bundles of Reductive Homogeneous Spaces
We

Intrinsic Rollings of Reductive Homogeneous Spaces
Let G/H be a reductive homogeneous space with fixed reductive decomposition g = h ⊕ m.
In the sequel, we always endow m with the covariant derivative ∇ m which is defined in In this section, we consider intrinsic (G(m)-reduced) rollings of (m, ∇ m ) over G/H equipped with an invariant covariant derivative ∇ α . Such intrinsic rollings are called rollings of m over G/H with respect to ∇ α , rollings of G/H with respect to ∇ α , or simply rollings of G/H, for short.
Notation 5.1 In the sequel, we do not explicitly refer to the G(m)-reduction if this reduction is clear by the context, for instance by denoting the configuration space by Q = m × (G × G(m)) as in Lemma 5.2, below.

Configuration Space
The goal of this subsection is to derive an explicit description of the configuration space for rollings of m over G/H with respect to an invariant covariant derivative ∇ α . Moreover, we consider a H-principal fiber bundle over the configuration space equipped with a suitable principal connection. This allows for lifting rollings, i.e. certain curves on the configuration space, horizontally to curves on that principal fiber bundle. We start with investigating the configuration space.
covering the identity id m×G/H : m × G/H → m × G/H whose inverse is given by 2. Let q = (v, [g, S]) ∈ Q with π(q) = (v, pr(g)). Then q defines the linear isomorphism via Lemma 3.1, Claim 2, where q is identified with Ψ −1 (q) ∈ (m × G(m)) × (G × H G(m)) /G(m). In the sequel, we often denote this isomorphism by q, as well, i.e. we write q(Z) = qZ = T g pr •T e ℓ g • S Z.
Proof: By Corollary 4.8 we have G(m, where pr is the canonical projection and Ψ is given by . Clearly, since (5.7) commutes and the canonical projection pr is a surjective submersion, the map Ψ defined by (5.4) is smooth by the smoothness of Ψ. In addition, Ψ maps fibers into fibers, i.e. it is a morphism of G(m)fiber bundles covering the identity of m × G/H. Therefore Ψ is an isomorphism of fiber bundles, see e.g. [5,Prop. 9.3]. The formula (5.5) for Ψ −1 is verified by a straightforward calculation. It remains to show Claim 2. Let q = (v, [g, S]) ∈ Q = m × (G × H G(m)) and let Z ∈ T v m. Then Ψ −1 ((v, [g, S])) = (v, id m ), [g, S] holds. Using the bijection from Lemma 3.1, Claim 2, this element is identified with a linear isomorphism which we denote by the same symbol. Evaluated at Z ∈ T v m ∼ = m, it is given by where the second equality follows by Lemma 3.1, Claim 2 and Corollary 4.8.

Remark 5.3
The configuration space π : Q → m×G/H can be viewed as a G(m)-principal fiber bundle. Indeed, as a consequence of Lemma 4.1, the G(m)-right action is a principal action.
Moreover, since the configuration space Q = m × (G × H G(m)) is the product of m and the associated bundle G × H G(m), we obtain an H-principal fiber bundle over Q. becomes a H-principal fiber bundle over Q = m × (G × H G(m)) with H-principal action given by where ⊳ : G × H ∋ (g, h) → g ⊳ h = gh ∈ G denotes the H-principal action from (2.42) on pr : G → G/H.

For
3. Let P ∈ Γ ∞ End(T G) and let ω ∈ Γ ∞ (T * G)⊗ h denote the principal connection and connection one-form from Proposition 2.20 on pr : G → G/H, respectively. Defining for (v, g, S) ∈ Q and (u, v g , v S ) ∈ T (v,g,S) Q yields a principal connection on π : Q → Q with corresponding connection one-form ω ∈ Γ ∞ (T * Q) ⊗ h given by (5.14) 4. Let q : I ∋ t → q(t) = (v(t), g(t), S(t)) ∈ Q be a horizontal curve with respect to the principal connection P. Then the curve g : I → G defined by the second component of q is horizontal with respect to Hor(G) from Proposition 2.20.
Proof: We consider pr : G → G/H as a H-principal fiber bundle. Then m × G ∋ (v, g) → (v, pr(g)) ∈ m × G/H becomes clearly a H-principal fiber bundle with principal action Thus, by the definition of an associated bundle, π : Q → Q becomes a H-principal fiber bundle over Q with principal action given by (5.10), i.e. Claim 1 is shown. Next, let P ∈ Γ ∞ End(T P ) be the principal connection on G from Proposition 2.20. It is straightforward to verify that P ∈ Γ ∞ End(T P ) defined by yields a principal connection on m × G → m × G/H with corresponding connection oneform given by ω (v,g) (u, v g ) = ω g (v g ). Thus Proposition 4.7 applied to m × G → m × G/H equipped with the principal connection P yields Claim 2, Claim 3, and Claim 4.

The Distribution characterizing Intrinsic Rollings
Motivated by [6, Sec. 4], we determine a distribution on Q characterizing intrinsic rollings of m over G/H. More precisely, a curve q : I → Q is horizontal with respect to this distributions iff it is a rolling of G/H with respect to ∇ α .
Applying the description of the tangent bundle of an associated bundle from (2.28) to the configuration space Q, we obtain for its tangent space Before we proceed, we state a simple lemma concerning this identification. We start with considering a situation which is slightly more general than (5.15).

holds.
Proof: We denote by ⊳ : G × H ∋ (g, h) → g ⊳ h = gh ∈ G the principal action on G → G/H. Its tangent map is given by due to (2.13). Moreover, the tangent map of where we identify T (H × G(V )) = T H × T G(V ). By setting v h = 0 in (5.17) and (5.18), respectively, we obtain T (g,h) (· ⊳ ·)(v g , 0) = T g r h v g and Thus the desired result follows by the definition of the equivalence relation in In order to determine the distribution on Q which characterizes rollings of m over G/H with respect to ∇ α , we first define a distribution on Q. Afterwards, this distribution is used to obtain the desired distribution on the configuration space Q.

Corollary 5.6 Let G/H be a reductive homogeneous space and let G(m) ⊆ GL(m) be a closed subgroup such that
Lemma 5.7 Let G/H be a reductive homogenoues space. Moreover, let G(m) ⊆ GL(m) be a closed subgroup and let g(m) ⊆ gl(m) denote its Lie algebra. Assume that Ad h m ∈ G(m) holds for all h ∈ H and let α : m × m → m be an Ad(H)-invariant bilinear map such that for each X ∈ m the linear map is an element in g(m), i.e. α(X, ·) ∈ g(m). Moreover, let Q = m × G × G(m) as in Proposition 5.4 and define Then Ψ α is a morphism of vector bundles covering id Q : Q → Q and D α = im(Ψ α ) ⊆ T Q is a regular distribution on Q given fiber-wise by for all (v, g, S) ∈ Q. Moreover, D α is contained in the the horizontal bundle defined by the principal connection P from Proposition 5.4, i.e.
is fulfilled.
Proof: The image of Ψ α defined by (5.21) is contained in T Q. Indeed, by the assumption on α : m × m → m, we have α(Su, ·) ∈ g(m) for S ∈ G(m) and u ∈ m. Hence we obtain Thus Ψ α is clearly a smooth vector bundle morphism covering the identity. Furthermore, the rank of Ψ α is obviously constant. Hence its image D α = im(Ψ α ) is a vector subbundle of T Q by [19,Thm. 10.34]. The fiber-wise description of D α in (5.22) holds by the definition of Ψ α due to D α = im(Ψ α ).
We now show that D α is contained in the horizontal bundle. Obviously, this is equiv- for some u ∈ m, we obtain due to v g = (T e ℓ g • S)u ∈ Hor(G) g because of Su ∈ m, where we used P g (v g ) = 0 as well as ω g (v g ) = 0 by the definitions of P ∈ Γ ∞ End(T G) and ω ∈ Γ ∞ (T * G) ⊗ h in Proposition 2.20.
Next we use the distribution D α on Q to construct the desired distribution on Q.
Lemma 5.8 Using the notations and assumptions of Lemma 5.7, we define D α ⊆ T Q by Then the following assertions are fulfilled: Then D α is fiber-wise given by using the identification (5.15) implicitly.
2. Let (v, g, S) ∈ Q. Then the map is a linear isomorphism.
3. Let q : I → Q be a curve and let q : I → Q denote a horizontal lift of q with respect to the principal connection from Proposition 5.4. Then q is horizontal with respect to D α , i.e.q(t) ∈ D α q(t) iff q is horizontal with respect to D α , i.e.q(t) ∈ D α q(t) . 4. D α is the image of the morphism of vector bundles over id Q : Q → Q of constant rank. In particular, D α is a regular distribution on Q.
Proof: We start with determining D α point-wise.
holds by the identification (5.15). Evaluating (5.28) for some u ∈ m, yields Claim 1 because of D α π(v,g,S) = (T π) D α (v,g,S) . Next we show Claim 2, i.e. that the restriction is bijective. Clearly, the linear map in (5.29) is injective since D α (v,g,S) ⊆ Hor(Q) (v,g,S) holds according to Lemma 5.7. We now show that (5.29) is surjective. Let h ∈ H. Moreover, let (v, g, S) ∈ Q and (v, gh, Ad h −1 •S) ∈ Q be two representatives of (5.33) By (5.32) and (5.33), we obtain due to Corollary 5.6. Thus the linear map (5.29) is surjective. Hence Claim 2 is proven. Next let q : I → Q be a curve and let q : I → Q be a horizontal lift with respect to the principal connection P from Proposition 5.4. In particular π(q(t)) = q(t) holds. Assume thatq(t) ∈ D α q(t) holds. This assumption yieldṡ by the definition of D α since q(t) is horizontal with respect to D α . Conversely, assume thatq(t) ∈ D α q(t) holds. Thenq(t) ∈ T q(t) Q is the unique horizontal tangent vector which fulfills (T q(t) π)q(t) =q(t) or equivalentlẏ q(t) = T q(t) π Hor(Q) q(t) −1q (t).
Since (5.29) is a linear isomorphism, we obtain T q(t) π D α q(t) −1 (D α q(t) ) = D α q(t) . This yields Claim 3 because ofq(t) ∈ D α q(t) and D α ⊆ Hor(Q). It remains to proof Claim 4. To this end, using the identification (5.15), we consider the diagram which clearly commutes. Thus Ψ α is smooth since T π • Ψ α is smooth and π × id m is a surjective submersion. In addition, Ψ α is fiber-wise linear, i.e. Ψ α is a vector bundle morphism covering id Q : Q → Q. Moreover, Claim 2 implies that the rank of Ψ is constant. Hence the image of Ψ α is a subbundle of T Q according to [19,Thm. 10.34]. In addition, we obtain D α = im(Ψ α ) due to belongs to g(m), i.e. to the Lie algebra of G(m). Let D α denote the distribution on Q = m × G × G(m) from Lemma 5.7 associated to α and let D α = T π(D α ) be the distribution defined in Lemma 5.8. Then the following assertions are fulfilled: 1. Let q : I → Q and let , S(t)) ∈ Q be a horizontal lift of q with respect to the principal connection P from Proposition 5.4. Then q is horizontal with respect to D α iff the ODEṠ is fulfilled. Moreover, the development curve is given by γ = pr •g : I → G/H.

2.
Let q : I → Q be a curve and let (v, γ) = (π • q) : I → m × G/H. Then q is horizontal with respect to D α , i.e.q(t) ∈ D α q(t) , iff q defines a (G(m)-reduced) intrinsic rolling of m over G/H with respect to ∇ α with rolling curve v and development curve γ.
Proof: We first show Claim 1. Let q : I → Q be some curve and let q : be a horizontal lift of q with respect to the principal connection P from Proposition 5.4. Clearly, the development curve γ : I → G/H defined by q : I → Q is given by γ = pr •g. Moreover, by Lemma 5.8, Claim 3, q is horizontal with respect to D α iff q is horizontal with respect to D α . Hence it is sufficient to show that q from (5.40) fulfills the ODE (5.39) iff q is horizontal with respect to D α . First we assumeq(t) ∈ D α q(t) for all t ∈ I. Writinġ q(t) = (v(t),ġ(t),Ṡ(t)) and using the definition of D α , one obtainṡ Thusġ(t) andṠ(t) are uniquely determined bẏ due to (5.41). Hence the curve q : I → Q which is horizontal with respect to D α fulfills the ODE (5.39). Conversely, assume that q : I → Q given by q(t) = (v(t), g(t), S(t)) fulfills (5.39). Then q(t) is clearly horizontal with respect to D α by the definition of D α . Thus Claim 1 is proven. Next we show Claim 2. To this end, let q : I → Q be horizontal with respect to D α . Then a horizontal lift q : I ∋ t → q(t) = (v(t), g(t), S(t)) ∈ Q of q fulfills the ODE (5.39) by Claim 1. Moreover, q : I → Q can be represented by Hence the linear isomorphism associated with q(t) is given by showing the no-slip condition. Next we prove the no-twist condition. Let be a vector field along v : I → m which we identify with the map I ∋ t → Z 2 (t) ∈ m defined by its second component. Then Z is parallel along v iffŻ 2 (t) = 0 holds, i.e. Z 2 (t) = Z 0 for all t ∈ I and some Z 0 ∈ m. We need to show that the vector field is parallel along γ : I → G/H with respect to ∇ α . By Proposition 5.4, Claim 4, the curve g : I → G is a horizontal lift of the curve γ : I ∋ t → pr(g(t)) ∈ G/H. In addition, we have by (5.39). Moreover, the horizontal lift of Z along g : I → G is given by Thus we obtain by exploiting (5.39) Hence Z is parallel along γ by Proposition 2.26. Conversely, assume that Z(t) = q(t)Z(t) is parallel along γ : I → G/H, where Z : I ∋ t → (v(t), Z 2 (t)) ∈ T m is some vector field along along v : I → m which we identify with the map I ∋ t → Z 2 (t) ∈ m. Let {A 1 , . . . , A N } ⊆ m be some basis of m. We define a parallel frame along γ : I → G/H by A i (t) = q(t)A i for i ∈ {1, . . . , N } and t ∈ I.
Then Z : [20,Chap. 4,p. 109]. By the linearity of q(t) : is a parallel vector field along the curve v : I → m. Thus the curve q : I → Q which is horizontal with respect to D α is a rolling.
It remains to proof the converse. Let q : I → Q be a curve defining a rolling. We show that q is horizontal with respect to D α .
Let q : I → Q be a horizontal lift of q with respect to the principal connection from Proposition 5.4. By Lemma 5.8, Claim 3 q is horizontal with respect to D α iff q is horizontal with respect to D α . Writing q(t) = (v(t), g(t), S(t)), the linear isomorphism T v(t) m ∼ = m → T pr(g(t)) (G/H) defined by q(t) = π(q(t)) is given by according to Lemma 5.2, Claim 2. Hence the no slip condition yieldṡ (5.43) By Proposition 5.4, Claim 3 the curve g : I → G is horizontal with respect to Hor(G) from Proposition 2.20. In addition, γ = pr •g holds, i.e. g : I → G is a horizontal lift of γ : I → G/H. Thus g : I → G fulfills the ODEġ(t) = T e ℓ g(t) • S(t) v(t) by (5.43). Moreover, since g : I → G is a horizontal lift of γ : I → G/H, the no twist condition yields for all Z 0 ∈ m by Proposition 2.26. Clearly, (5.44) is equivalent to the ODĖ for S : I → G(m). Thus q : I → Q is horizontal with respect to D α by Claim 1. Therefore q : I → Q is horizontal with respect to D α due to Lemma 5.8, Claim 3.
In particular, Theorem 5.9 applies to (pseudo-)Riemannian reductive homogeneous spaces. We comment on this particular situation in the next remark. Remark 5.10 can be specialized further to naturally reductive homogeneous space equipped with the Levi-Civita covariant derivative.

Remark 5.11
Let G/H be a naturally reductive homogeneous space. Then Theorem 5.9 can be applied to G/H equipped with ∇ LC , where the configuration space can be reduced to Q = m × (G × H O(m)) and α : m × m → m is given by α(X, Y ) = 1 2 pr m • ad X (Y ) for X, Y ∈ m since ∇ LC = ∇ can1 holds by Remark 2.30.

Kinematic Equations and Control Theoretic Perspective
Throughout this section we denote by G/H a reductive homogeneous space and we assume that G(m) ⊆ GL(m) is a closed subgroup such that Ad h m ∈ G(m) holds. Moreover, we assume that the Ad(H)-invariant bilinear map α : m × m → m fulfills α(X, ·) ∈ g(m) for all X ∈ m. If not indicated otherwise, we consider the "reduced" configuration space Q = m × (G × H G(m)).
We start with relating rollings of m over G/H to a control system. Then Ψ α defines a control system in the sense of [13, p. 21] with state space Q and control set m. Obviously, for each u ∈ m, the map Ψ α (·, u) : Q → T Q is a section of D α , where D α ⊆ T Q is the distribution characterizing the rolling of m over G/H with respect to ∇ α .
Moreover, if G/H is equipped with an invariant pseudo-Riemannian metric and an invariant metric covariant derivative ∇ α , we can endow Q with an additional structure which is similar to a sub-Riemannian structure. We refer to [1,Def. 3.2] for a definition of sub-Riemannian structures.
Then the pair (Ψ α , Q × m) is formally similar to a sub-Riemannian structure on Q except for the following facts: 1. The fiber metric on Q × m → m is allowed to be indefinite.
2. In general, the manifold Q is not connected.
However, by imposing further restrictions on G/H, Q and ·, · it might be possible to obtain a sub-Riemannian structure on Q. In particular, if we assume that G/H is a Riemannian reductive homogeneous space, the fiber metric h on Q is positive definite. Moreover, if we assume that G is connected and Ad h m : m → m is an orientation preserving isometry, i.e. Ad h m ∈ SO(m) for all h ∈ H, the configuration space can be reduced to Q = m × (G × H SO(m)), which is obviously connected. Under these assumptions, the pair (Ψ α , Q × m) defines a structure on Q which fulfills the requirements of a sub-Riemannian structure on Q in the sense of [1, Def. 3.2] except for the fact that D α might be not bracket generating. Investigating conditions on G/H and α such that D α is bracket generating is out of the scope of this text. Nevertheless, in this context, we refer to [7], where the controllability of rollings of oriented Riemannian manifolds are considered. Moreover, we mention [14], where optimal control problems associated to rollings of certain manifolds are considered.
Using terminologies of control theory, we call a curve u : I → m a control curve. Such a curve can be used to determine a rolling of m over G/H, where the rolling curve v : I → m satisfies the ODEv(t) = u(t). Inspired by the terminology used in [12], we introduce a notion of a kinematic equation for rollings of m over G/H with respect to ∇ α . To this end, we first state the following proposition.

(5.46)
Then q : I ∋ t → (π • q)(t) ∈ Q is a rolling of m over G/H with respect to ∇ α .
Proof: The curve q : I ∋ t → (π • q)(t) = (v(t), [g(t), S(t)]) ∈ Q is horizontal with respect to D α ⊆ T Q by Theorem 5.9, Claim 1 because of u(t) =v(t) for all t ∈ I. Hence q : I → Q is a rolling of m over G/H by Theorem 5.9, Claim 2.

Definition 5.15
The ODE (5.46) in Proposition 5.14 is called the kinematic equation for (G(m)-reduced) rollings of m over G/H with respect to ∇ α . An initial value problem associated with the ODE (5.46) with some initial condition (v(t 0 ), g(t 0 ), S(t 0 )) ∈ m × G × G(m) for some t 0 ∈ I is called kinematic equation, as well.

Remark 5.16
By specializing ∇ α in Definition 5.15, one obtains: 1. The kinematic equation with respect to ∇ can1 readṡ v(t) = u(t), for all t ∈ I and some t 0 ∈ I. We point out that setting S(t) = id m for all t ∈ I yields an expression which is similar to the ODE describing rollings of a symmetric space over a flat space obtained [15,Sec. 4.2].
Next, we state the kinematic equation for a naturally reductive homogeneous space. by [22,Prop. 23.9]. We view O(m) as subset of End(m) and denote by ·, · : m × m → m the Ad(H)-invariant inner product corresponding to the Riemannian metric on G/H. The norm on m induced by ·, · is denoted by · . We denote an extension of these maps to g by the same symbols. We now endow End(m) with the Frobenius scalar product given by S, T F = tr(S ⊤ T ), where S ⊤ is the adjoint of S with respect to ·, · . Then ·, · F induces a bi-invariant and hence a complete Riemannian metric on O(m). Moreover, the norm · F defined by the Frobenius scalar product is equivalent to the operator norm · 2 . In particular, there is a C > 0 such that S F ≤ C S 2 holds for all S ∈ End(m).
We now endow Ê × O(m) with the Riemannian metric defined for (s, S) ∈ Ê × O(m) and which is clearly complete. Moreover, α : m×m → m is bounded since m is finite dimensional. Hence there exists a C ′ ≥ 0 with α(X, Y ) ≤ C ′ X Y for X, Y ∈ m. Thus, for fixed X ∈ m, the operator norm of the linear map α(X, ·) : m → m is bounded by (5.52) By this notation, we obtain is defined on Ê.

Rolling along Special Curves
Next we consider a rolling of m over G/H along a curve such that the development curve γ : I → G/H is the projection of a not necessarily horizontal one-parameter subgroup of G, i.e. γ(t) = pr(exp(tξ)), t ∈ I (5.55) for some ξ ∈ g. In this subsection, we focus on the invariant covariant derivatives ∇ can1 and ∇ can2 on G/H. This discussion is motivated by the rolling and unwrapping technique for solving interpolation problems, see e.g. [10,12], for which an explicit expression for a rolling along a curve connecting two points is desirable. A natural choice for such a curve would be a projection of a horizontal one-parameter subgroup in G, i.e. a geodesic with respect to ∇ can1 or ∇ can2 . However, even if such a curve connecting two given points exists, as far as we know, in general, no closed-formula for such curves are known. In this context, we refer to [18], where the problem of connecting two points X, X 1 ∈ St n,k ⊆ Ê n×k on the Stiefel manifold St n,k by a curve of the form t → e tξ 1 Xe tξ 2 with some suitable (ξ 1 , ξ 2 ) ∈ so(n) × so(k) is addressed. As preparation for deriving the desired rollings, we state the following lemma as preparation.

Lemma 5.19
Let G/H be a reductive homogeneous space with reductive decomposition g = h ⊕ m, let ξ ∈ g and let γ : I ∋ t → pr(exp(tξ)) ∈ G/H. Then the curve is the horizontal lift of γ through g(0) = e with respect to the principal connection from Proposition 2.20. Moreover, g is the solution of the initial value probleṁ Proof: Obviously, γ(t) = pr(exp(tξ)) = pr(g(t)) holds for all t ∈ I due to exp(−tξ h ) ∈ H for all t ∈ I. Moreover, g(0) = e is fulfilled. It remains to prove that g : I → G is horizontal.
To this end, we compute by exploiting (2.13) and (2.12) Consequently, we obtain by the chain rule Since G/H is reductive, this implies (T e ℓ g(t) ) −1ġ (t) = Ad exp(tξ h ) (ξ m ) ∈ m (5.58) due to exp(tξ h ) ∈ H. Thus g is horizontal. Moreover, the curve g : I → G is a solution of the initial value problem (5.57) by (5.58).

Remark 5.20
Let G be equipped with a bi-invariant metric which induces a positive definite fiber metric on Hor(G), i.e. a sub-Riemannian structure on G, and let H ⊆ G be a closed subgroup such that its Lie algebra h ⊆ g is non-degenerated with respect to the scalar product corresponding to the metric. Then the curve given by g(t) = exp(tξ) exp(−tξ h ) from Lemma 5.19 is a sub-Riemannian geodesic on G according to [23,Sec. 11.3.7].

Rolling along Special Curves with respect to ∇ can1
We now derive an expression for a rolling of m over G/H with respect to ∇ can1 such that the development curve is given by γ : I ∋ t → pr(exp(tξ)) ∈ G/H for some ξ ∈ g. To this end, we determine a curve q : which is horizontal with respect to the principal connection P from Proposition 5.4, Claim 3 such that q = π • q : I → Q is the desired rolling. In particular, pr(g(t)) = γ(t) = pr(exp(tξ)) (5.60) has to be fulfilled and q has to be tangent to the distribution D α by Theorem 5.9, Claim 2. Thus q : I → Q has to be tangent to D α by Lemma 5.8, Claim 3. Furthermore, by Proposition 5.4, Claim 4, the curve g : I → G fulfilling γ = pr •g is tangent to Hor(G) ⊆ T G from Proposition 2.20, i.e. g is a horizontal lift of γ. Hence Lemma 5.19 yields for all t ∈ I which fulfills the initial value probleṁ In order to obtain the desired rolling, we need to solve the initial value problem associated with (5.65) explicitly. This is the next lemma. Proof: We make the following Ansatz. We set S(t) = Ad exp(tξ h ) • S(t) for all t ∈ I, where S : I → O(m) is given by S(t) = exp − t(ad ξ h + 1 2 (pr m • ad ξm )) • S 0 for t ∈ I and some S 0 ∈ O(m). Obviously, holds for all t ∈ I. Using the well-known identity Ad exp(tξ h ) = e t ad ξ h , we compute Thus we obtaiṅ where we exploited that pr m and Ad h commutes and that Ad h is a morphism of Lie algebras for all h ∈ H. Moreover, S(0) = S(0) = S 0 is clearly fulfilled. Hence Thus the desired rolling is determined. The discussion above is summarized in the next proposition.

Proposition 5.22
Let G/H be a reductive homogeneous space with reductive decomposition g = h⊕m and let ξ ∈ g be arbitrary. Moreover, let q : for t ∈ I. Then q : Furthermore, this intrinsic rolling viewed as a triple as in Remark 3.3 is given by (v(t), γ(t), A(t)), where A(t) reads Proof: This is a consequent of the above discussion. Then q : I ∋ t → q(t) = (v(t), [g(t), S(t)]) ∈ Q is an intrinsic rolling with respect to ∇ can1 whose development curve is a geodesic with respect to ∇ can1 .
Proof: This is an immediate consequence of Proposition 5.22 due to ξ h = 0.

Rolling along Special Curves with respect to ∇ can2
We now consider a rolling of a reductive homogeneous space with respect to the covariant derivative ∇ can2 such that the development curve is given by γ : I ∋ t → pr(exp(tξ)) for some ξ ∈ g. This is the next proposition.
Proposition 5.25 Let ξ ∈ g be arbitrary and define q :  Then q = π • q : I → Q is an intrinsic rolling of m over G/H with respect to ∇ can2 whose development curve is given by γ : I ∋ t → pr(exp(tξ)) ∈ G/H. This intrinsic rolling viewed as a triple as in Remark 3.3 is given by (v(t), γ(t), A(t)), where A(t) reads (5.77) Proof: The curve g : I ∋ t → exp(tξ) exp(−tξ h ) ∈ G is the horizontal lift of γ through g(0) = e by Lemma 5. 19. We now show that q : I → Q defined by (5.76) fulfills the kinematic equation from Remark 5.16, Claim 2. Indeed, we have and g(t) = exp(tξ) exp(−tξ h ) is the solution of the initial value probleṁ by Lemma 5.19 as desired. Therefore q = π • q : I → Q is indeed a rolling of m over G/H with respect to ∇ can2 whose developement curve is given by γ(t) = pr(exp(tξ)).

Applications and Examples
In this section, we consider some examples. Before we study Lie groups and Stiefel manifolds in detail, we briefly comment on symmetric homogeneous spaces. By recalling ∇ can1 = ∇ can2 for symmetric homogeneous space and ∇ LC = ∇ can1 = ∇ can2 for pseudo-Riemannian symmetric homogeneous spaces from Remark 2.31, the kinematic equation from Remark 5.16, Claim 2 yields the next lemma.

Rolling Lie Groups
In this subsection, we discuss intrinsic rollings of Lie groups. First we discuss rollings of g over G, where we view G as the reductive homogeneous space G/{e} equipped with the covariant derivative ∇ can1 . Afterwards, we discuss rollings of a connected Lie group G viewed as the symmetric homogeneous space (G × G)/∆G equipped with ∇ can1 = ∇ can2 . It turns out that both points of view are closely related.

Rollings of Lie Groups as Reductive Homogeneous Spaces
We first consider the rolling of a Lie-group G viewed as a reductive homogeneous space G/{e} equipped with the covariant derivative ∇ can1 . Obviously, the reductive decomposition is given by h = {0} and m = g. Clearly, this implies pr m = id m = id g . Moreover, the configuration space becomes We now determine a rolling q : To this end, we solve the following initial value problem associated with the kinematic equation from Remark 5.16, Claim 1 v(t) = u(t), v(0) = 0, where u : I → g denotes a prescribed control curve. Motivated by [12,Sec. 3.2], where rollings of SO(n) over one if its affine tangent spaces are determined by using an extrinsic point of view, we make the following Ansatz.
We define the curves k : I → G and W : I → G by the initial value problemṡ Moreover, we set S : I → GL(g), t → S(t) = Ad W (t) (6.6) as well as g : I → G, t → g(t) = k(t)W (t) −1 . (6.7) Clearly, S(0) = Ad e = id g and g(0) = g 0 e −1 = g 0 holds. Next we show that S : I → GL(g) defined by (6.6) is a solution of (6.4). To this end, we calculatė where we used the chain-rule and exploited the definition of W in (6.5). In other words, the smooth curve fulfills γ(0) = W (t) and d ds γ(s) s=0 =Ẇ (t) according to 6.8. Thus we calculate for Z ∈ g by using the definition of S : I → GL(g) from (6.6) and the chain rulė as desired, where we exploited that Ad g : g → g is a morphism of Lie algebras for all g ∈ G and γ : Ê → G is defined by (6.9).
It remains to show that g : I → G defined in (6.7) fulfills (6.4). To this end, using the chain-rule several times, we obtain by g(t) = k(t)W (t) and (2.13) as well as (2.14) and the definition of k : I → G and W : I → G in (6.5) = T e ℓ g(t) • S(t) u(t) (6.11) as desired. Hence q : is an intrinsic rolling of g over G/H with respect to ∇ can1 , where S and g are defined in (6.6) and (6.7), respectively. Moreover, v : I → g is determined byv(t) = u(t) and the initial value v(0) = 0. We summarize the above discussion in the next proposition.
Proposition 6.2 Let G be a Lie group viewed as reductive homogeneous space G/{e} equipped with ∇ can1 . Let u : I → g be some control curve and define k : I → G as well as W : I → G by the initial value problemṡ k(t) = 1 2 T e ℓ k(t) u(t), k(0) = g 0 andẆ (t) = − 1 2 T e ℓ W (t) u(t), W (0) = e. (6.13) Then q : I ∋ t → (v(t), g(t), S(t)) = v(t), k(t)W (t) −1 , Ad W (t) ∈ g × G × GL(g) = Q (6.14) is an intrinsic rolling of g over G, where the development curve v : I → g is defined by v(t) = t 0 u(s) ds (6.15) and the rolling curve is given by g : This rolling can be viewed as a triple (v(t), g(t), A(t)) as in Remark 3.3, where the linear isomorphism A(t) : T v(t) g ∼ = g → T g(t) G is given by for all Z ∈ g.
Remark 6.3 Let u : I → g be a control curve. Then the intrinsic rolling q : I → m × G × GL(g) of g over G with respect to ∇ can1 is defined on the whole interval I by the form of the initial value problem in (6.13).
Corollary 6.4 Let G be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. Then rollings of g over G with respect to ∇ LC with a prescribed control curve u : I → g are given by Proposition 6.2.
Proof: Let G be equipped with a pseudo-Riemannian bi-invariant metric. Then the corresponding scalar product ·, · : g × g → Ê is Ad(G)-invariant, see e.g. [ [3]. The tangential part of these rollings is very similar to the result of Proposition 6.2. Indeed, the linear isomorphism defined by the rolling from Proposition 6.2 in (6.16) simplifies for a matrix Lie group to for all Z ∈ g.

Rollings of Lie Groups as Symmetric Homogeneous Spaces
We now identify G with the symmetric homogeneous spaces (G × G)/∆G and study the rolling of m over (G × G)/∆G with respect to ∇ can1 = ∇ can2 . To this end, we state the next lemma as preparation which is an adaption of [4, Sec. 23.9.5], see also [8,Chap. IV,6], where it is stated for the Riemannian case.

(6.18)
Then σ is an involutive automorphism of G × G and ∆G = {(g, g) | g ∈ G} ⊆ G × G is the set of fixed points of σ. Moreover, (G × G)/∆G is a symmetric homogeneous space and the corresponding canonical reductive decomposition g × g = h ⊕ m is given by h = {(X, X) | X ∈ g} and m = {(X, −X) | X ∈ g}. (6.19) In addition, the map is a diffeomorphism and the map is a surjective submersion which fulfills φ • pr = φ.
Next we determine the tangent map of φ evaluated at elements in Hor(G × G) ⊆ T G × T G. We point out that the identity T (e,e) φ (X, X) = 2X for all X ∈ g is wellknown, see e.g. [4,Sec. 23.9.5] or [8,Chap. IV,6]. Lemma 6.7 Let G be a connected Lie group and let X ∈ g. Then (T e ℓ g 1 X, −T e ℓ g 2 X) ∈ Hor(G × G) (g 1 ,g 2 ) holds. Moreover, the tangent map of φ : • Ad g 2 (2X).

(6.22)
In particular T (e,e) φ(X, −X) = 2X holds and is satisfied for all X ∈ g.
Next we consider intrinsic rollings of m over (G × G)/∆G with respect to ∇ can2 and relate them to the intrinsic rollings of g over G with respect to ∇ can1 . This is the next proposition.

Rolling Stiefel Manifolds
Rollings of Stiefel manifolds have been already considered in the literature in [9] and [15], however not from an intrinsic point of view. In this section we apply the general theory developed in Section 5 to the Stiefel manifold St n,k endowed with the Levi-Civita covariant derivative defined by a so-called α-metric. These metrics have been recently introduce in [11]. Remark 6. 9 We point out that in contrast to the previous sections, where α denotes a bilinear map m×m → m defining an invariant covariant derivative, in this section α denotes an element in Ê\{0}. There is no danger of confusion because in this section, we consider rollings of Stiefel manifolds exclusively with respect to the Levi-Civita covariant derivative ∇ LC defined by an α-metric. Since the Stiefel manifold St n,k equipped with an α-metric is a naturally reductive homogeneous space, see Lemma 6.11 below, the Levi-Civita covariant derivative ∇ LC corresponds to the invariant covariant derivative defined by the bilinear map m × m ∋ (X, Y ) → 1 2 [X, Y ] m ∈ m according to Remark 2.30.
Definition 6.10 Let α ∈ Ê\{0,−1} and let O(n)× O(k) be equipped with the bi-invariant metric defined by the scalar product ·, · α from (6.38). The metric on St n,k defined by requiring that the map pr X : O(n) × O(k) → St n,k from (6.35) is a pseudo-Riemannian submersion is called α-metric.

Intrinsic Rolling
We now determine intrinsic rollings of the Stiefel manifold equipped with an α-metric over one of its tangent spaces. By Lemma 5.2, the configuration space for rolling T X St n,k ∼ = m over St n,k intrinsically is given by the fiber bundle where pr m : so(n) × so(k) → m is explicitly given by Lemma 6.12. Let q : I ∋ t → (v(t), g(t), S(t)) ∈ Q = m × (O(n) × O(k)) × O(m) be a curve satisfying (6.47). Then q : I → Q, t → q(t) = (π • q)(t) = (v(t), [g(t), S(t)]) (6.48) is an intrinsic rolling of T X St n,k ∼ = m over St n,k with respect to the given α-metric along the rolling curve v. The development curve I ∋ t → pr(g(t)) = (R(t), θ(t)) ∈ O(n) × O(k) is mapped by the embedding ι X : (O(n) × O(k))/H → Ê n×k to the curve γ : I → St n,k , t → γ(t) = (ι X • pr)(g(t)) = pr X (g(t)) = R(t)Xθ(t) ⊤ . (6.49) Proof: Since St n,k equipped with an α-metric is a naturally reductive homogeneous space by Lemma 6.11, this is a direct consequence of Corollary 5.17 combined with Lemma 6.12 and Lemma 6.13.
Next we consider the intrinsic rolling of the Stiefel manifolds along curves of a special form by using Section 5.4.1. This yields the next remark.

Conclusion
In this text, we investigated intrinsic rollings of reductive homogeneous spaces equipped with invariant covariant derivatives. As preparation, we considered frame bundles of vector bundles associated to principal fiber bundles in detail. Afterwards, using an abstract definition of intrinsic rolling as starting point, we investigated rollings of m over the reductive homogeneous spaces G/H with respect to an invariant covariant derivative ∇ α . For a given control curve, we obtained the so-called kinematic equation which is a time-variant explicit ODE on a Lie group, whose solutions describe rollings of m over G/H. Moreover, for the case, where the development curve is the projection of a one-parameter subgroup, we provided explicit solutions of the kinematic equation describing intrinsic rollings of m over G/H with respect to the canonical covariant derivative of first kind and second kind, respectively. As examples, we discussed intrinsic rollings of Lie groups and Stiefel manifolds.