A geometric approach to discrete connections on principal bundles

This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and relationships. An existence result for discrete connections on principal bundles equipped with appropriate Riemannian metrics is proved.


Introduction
The study of symmetries is a central part of many areas of Mathematics and Physics. In the Differential Geometric setting, principal bundles provide a powerful instrument to model many symmetric systems. Connections on principal bundles are very convenient tools, especially in the topologically nontrivial case. Among other things, using connections, the equations of motion of mechanical systems can be written globally and, also, connections capture the complexity of the bundle via, for instance, the associated curvature.
Roughly speaking, mechanical systems are continuous time dynamical systems on the tangent bundle T Q of a manifold Q, whose dynamics is defined using a variational principle. In the same spirit, discrete mechanical systems are usually introduced as discrete time dynamical systems on Q × Q whose dynamics is defined using a variational principle [11]. The main motivation is that, for continuous time, one has velocities (tangent vectors), whereas when the time is discrete, one has pairs of points (close to one another).
Let G be a Lie group acting on Q via the (left) action l Q : G × Q → Q in such a way that the quotient mapping π : Q → Q/G is a principal G-bundle. The vertical bundle V is defined, at each q ∈ Q, by V(q) := T q l Q G (q) = {ξ Q (q) : ξ ∈ Lie(G)} ⊂ T q Q, where l Q G (q) is the G-orbit through q and ξ Q (q) is the infinitesimal generator of l Q at q. A connection A on the principal G-bundle π is a G-equivariant distribution Hor A on Q complementing V. That is, at each q ∈ Q, there is a subspace Hor A (q) ⊂ T q Q such that, for each v q ∈ T q Q, there is a unique decomposition .
Trying to extend techniques that were used to analyze the reduction of symmetric mechanical systems ([2] and [1]) to the case of discrete mechanical systems, M. Leok, J. Marsden and A. Weinstein introduced in [9] and [8] a notion of discrete connection on a principal bundle that mimics the notion of connection. Their definition states that a discrete connection A d is a G-equivariant subset Hor A d ⊂ Q × Q that is complementary to the discrete vertical bundle V d := {(q, l Q g (q)) ∈ Q × Q : g ∈ G} where, for g ∈ G, l Q g : Q → Q is defined by l Q g (q) := l Q (g, q). Complementary means that every (q 0 , q 1 ) ∈ Q × Q can be decomposed uniquely in a vertical part and a horizontal part in such a way that (1.2) (q 0 , q 1 ) = (q 0 , l Q g (q 0 )) ∈V d · (q 0 , l Q g −1 (q 1 )) ∈HorA d for some g ∈ G. The composition of vertical and arbitrary pairs (based at the same point q 0 ) is defined by (q 0 , l Q g (q 0 )) · (q 0 , q 1 ) := (q 0 , l Q g (q 1 )). The basic intuition is that tangent vectors in T Q become finite curves and, eventually, pairs of points, elements of Q × Q. Vertical vectors based at q ∈ Q are those tangent vectors pointing in the direction of the group action which, for finite time, leads to pairs of the form (q, l Q g (q)) for g ∈ G. With this motivation, (1.2) is the discrete analogue of (1.1).
Discrete connections have been successfully used to study the reduction of discrete mechanical systems (see [10], [4], and [3]).
In their work, Leok, Marsden and Weinstein do not provide a thorough definition of discrete connection, although they discuss the equivalence between this notion and other approaches. For instance, they relate a discrete connection to what they call a discrete connection form and also to a discrete horizontal lift. They also give an interpretation in terms of splittings of a certain discrete Atiyah sequence, although the groupoid setting for this very intriguing approach is not detailed. The purpose of the current paper is to give a precise definition of discrete connection on a principal bundle and analyze some of its more basic properties. Additional geometric properties like parallel transport, holonomy and curvature will be discussed elsewhere.
In Section 2 we define discrete connections and study some elements associated to them: domain and slices. We can associate other objects to a discrete connections A d that, in turn, can be used to characterize completely A d . Two such objects are the discrete connection form and the discrete horizontal lift, that are analyzed in Sections 3 and 4 respectively. Last, in Section 5, we prove an existence result for discrete connections on principal bundles equipped with an adequate Riemannian metric.
Notation: when l Q is the left G-action on Q, l Q×Q is the induced diagonal G-action on Q × Q. The quotient mapping from a space to its space of orbits is denoted by π and p j is the projection from a Cartesian product onto its j-th factor. Given the maps f i : X i → Y i (for i = 1, 2), their Cartesian product is

Geometric definition
In order to use discrete connections in geometry, it is important that they are well defined and that the objects that we associate to them (discrete connection form, horizontal lift, etc.) be smooth.
Definition 2.1. Let Hor ⊂ Q × Q be an l Q×Q -invariant submanifold containing the diagonal ∆ Q ⊂ Q × Q. We say that Hor defines the discrete connection A d on the principal bundle π : Q → Q/G if (id Q × π)| Hor : Hor → Q × (Q/G) is an injective local diffeomorphism. We denote Hor by Hor A d .
Remark 2.2. It is possible to consider the slightly more general notion of affine discrete connection that replaces the condition ∆ Q ⊂ Hor with the requirement that Hor contains the graph of a smooth map γ : Q → Q. Discrete connections correspond to the case γ := id Q . This more general notion has been used in [4] in order to construct discrete connections associated to nonvanishing conserved discrete momenta of discrete mechanical systems.
The following Lemma, whose proof is straightforward, provides a convenient way to characterize discrete connections.
Lemma 2.3. The requirement that (id Q × π)| Hor : Hor → Q × (Q/G) be an injective local diffeomorphism in Definition 2.1 is equivalent to the following two assertions being true simultaneously.
(1) (id Q × π)| Hor : Hor → Q × (Q/G) is a local diffeomorphism and (2) l Q×Q2 g (Hor)∩Hor = ∅ for all g = e in G. Here l Q×Q2 g (q 0 , q 1 ) := (q 0 , l Q g (q 1 )). Recall that a smooth map f : for all x ∈ f −1 (Z); this situation is denoted by f ⊤ ∩ Z. When f ⊤ ∩ Z and f −1 (Z) = ∅, f −1 (Z) ⊂ X is a regular submanifold (see Theorem on page 28 of [5]). When Z 1 , Z 2 ⊂ Y are submanifolds, they intersect transversely when i Z1 ⊤ ∩ Z 2 , where i Z1 : Z 1 → Y is the inclusion map; this is denoted by Z 1 ⊤ ∩ Z 2 . In particular, if not empty, Z 1 ∩ Z 2 ⊂ Y is a submanifold when Z 1 ⊤ ∩ Z 2 . We refer to [5] for more on transversality. For any q ∈ Q define the smooth map i q : Q → Q × Q by i q (q ′ ) = (q, q ′ ). The next result proves the basic properties of a discrete connection.
Proposition 2.4. Let Hor A d be a discrete connection on the principal G-bundle π : Q → Q/G. Then, the following statements are true. (2) U is G × G-invariant for the product action on Q × Q. Also, U = (π × π) −1 (U ′′ ). (3) For any q ∈ Q, i q ⊤ ∩ Hor A d . Furthermore, Hor 2 (q) : (4) For any q ∈ Q, Hor 2 (q) ⊤ ∩ V d (q), where V d (q) := l Q G (q). More precisely, Proof. Being A d a discrete connection, U ′ is open and, as U := l Q×Q2 U is also open. Being π a fiber bundle, it is an open map and so is π × π; consequently, (π × π)(U) = (π × π)(Hor A d ) = U ′′ is open, proving part 1. The G × G-invariance of U follows from the G-invariance of Hor A d by direct computation. This G×G-invariance together with (π×π)(U) = U ′′ lead to the proof of point 2. Let where the first term is in T (q,q ′ ) Hor A d and the second in Im(di q (q ′ ))). This proves the transversality condition in point 3. The rest of this point is a consequence of the transversality condition (see Theorem on page 28 of [5]) and the fact that , proving the transversality part of point 4. The direct sum property follows from the dimensions of the subspaces.
Given a discrete connection A d , the open subset U ⊂ Q × Q defined in part 1 of Proposition 2.4 will be called the domain of A d . The submanifolds Hor 2 (q) ⊂ Q introduced in part 3 of the same result will be the horizontal slices.
Proposition 2.5. Let A d be a discrete connection with domain U on the principal G-bundle π : Q → Q/G. For any (q 0 , q 1 ) ∈ U, there is a unique g ∈ G such that (1.2) holds.
Proof. The existence of g follows from the definitions of U and the composition ·. The uniqueness of g is a consequence of (id Q × π)| HorA d being a injective.

Discrete connection form
A convenient way of describing and using a connection on a principal G-bundle π : Q → Q/G is through the associated connection 1-form, that is a Lie algebra valued 1-form A : [7] for more details). In the same spirit, when A d is a discrete connection on the same bundle, the element g ∈ G in (1.2), captures the vertical part of a pair (q 0 , q 1 ) in the sense that "what is left" is horizontal. The next definition makes this notion more precise.
Definition 3.1. Given a discrete connection A d with domain U on the principal G-bundle π : Q → Q/G, we define its associated discrete connection form where g is the element of G that appears in the decomposition (1.2).
When π : Q → Q/G is a principal G-bundle, the fibered product of π with itself -that is, the pairs (q 0 , q 1 ) such that π(q 0 ) = π(q 1 )-is denoted by Q × π×π Q. Let κ : Q × π×π Q → G be defined by κ(q 0 , q 1 ) := g if and only if l Q g (q 0 ) = q 1 . It is easy to check that κ is a smooth function.
Example 3.3. Let R be a smooth connected manifold and G a Lie group. Define Q := R × G and consider the left G-action on Q defined by l Q g (r, g ′ ) := (r, gg ′ ). This action turns Q into the principal G-bundle p 1 : Q → R. Let U ′′ ⊂ R × R be an open subset containing the diagonal ∆ R and C : U ′′ → G be a smooth function such that C(r 0 , r 0 ) = e for all r 0 ∈ R. Consider Hor ⊂ Q × Q is a regular submanifold because Hor is the graph of the smooth map ((r 0 , g 0 ), r 1 ) → ((r 0 , g 0 ), (r 1 , g 0 (C(r 0 , r 1 )) −1 )). It is immediate that Hor is l Q×Qinvariant and contains ∆ Q . Furthermore, as (id Q × π)| Hor : Hor → Q × (Q/G) specializes to ((r 0 , g 0 ), (r 1 , g 0 (C(r 0 , r 1 )) −1 )) → ((r 0 , g 0 ), r 1 ) that is a diffeomorphism with inverse ((r 0 , g 0 ), r 1 ) → ((r 0 , g 0 ), (r 1 , g 0 (C(r 0 , r 1 )) −1 )), we conclude that Hor defines a discrete connection A C d on the (trivial) principal In the special case when U ′′ = R × R and C(r 0 , r 1 ) = e for all r 0 , r 1 ∈ R, the connection A e d is called the trivial discrete connection.
Given (q 0 , q 1 ) = ((r 0 , g 0 ), (r 1 , g 1 )) ∈ U, it is easy to see that (q 0 , l Q g −1 (q 1 )) ∈ Hor A C d if and only if g = g 1 C(r 0 , r 1 )g −1 0 . Therefore, the associated discrete connection form is Then, for all (q 0 , q 1 ) ∈ U and g 0 , g 1 ∈ G, is an open subset that contains the diagonal ∆ Q ⊂ Q × Q and is invariant under the product G × G-action on Q × Q, such that (3.4) holds (with A d replaced by A) and A(q 0 , q 0 ) = e for all q 0 ∈ Q, then Hor := {(q 0 , q 1 ) ∈ U : A(q 0 , q 1 ) = e} defines a discrete connection whose associated discrete connection form is A.
Proof. For (q 0 , q 1 ) ∈ U, let h := A d (q 0 , q 1 ). By the G × G-invariance of U, we have that (l Q g0 (q 0 ), l Q g1 (q 1 )) ∈ U for any g 0 , g 1 ∈ G; let h := A d (l Q g0 (q 0 ), l Q g1 (q 1 )). By definition, (q 0 , l Q h −1 (q 1 )) ∈ Hor A d and by the G-invariance of Hor A d , we see On the other hand, also by definition, Using that (id Q × π)| HorA d is one to one, we conclude that g 1 hg −1 0 = h, proving that (3.4) holds. By Proposition 2.5, the element g appearing in (1.2) is unique, hence g = e characterizes the horizontality of (q 0 , q 1 ) ∈ U. This proves that Conversely, given U, A and Hor as in the statement, we show that Hor defines a discrete connection. Since A(q 0 , q 0 ) = e for all q 0 ∈ Q, we have that ∆ Q ⊂ Hor. It is easy to check explicitly that dA : T (q0,q1) U → T e G = g is onto for all (q 0 , q 1 ) ∈ Hor. Hence, e is a regular value of A and Hor := A −1 ({e}) ⊂ Q × Q is a submanifold. The l Q×Q -invariance of Hor follows readily using (3.4).
Assume now that (q 0 , l Q g (q 1 )) ∈ Hor with (q 0 , q 1 ) ∈ Hor. Then e = A(q 0 , l q g (q 1 )) = gA(q 0 , q 1 ) = ge = g, showing that condition 2 in Lemma 2.3 holds. That (id Q × π)| Hor is a local diffeomorphism is checked locally. Let U ⊂ Q/G be an open subset trivializing π and σ ∈ Γ(U, Q| U ), a local section over U of the principal G-bundle π : ). It can be checked that Φ σ is a smooth map whose image is contained in Hor. Furthermore, Φ σ is the inverse of (id Q × π)| Hor∩((Q|U )×(Q|U )) , showing that condition 1 in Lemma 2.3 holds. By Lemma 2.3, (id Q × π)| Hor is an injective local diffeomorphism and we conclude that Hor defines a discrete connection A d on π : Q → Q/G. Direct evaluation shows that the domain of A d is U = U and that the discrete connection form associated to A d is A.
Motivated by the previous analysis we introduce the following concept.
Definition 3.5. Let π : Q → Q/G be a principal G-bundle and U ⊂ Q × Q be an open subset that contains the diagonal ∆ Q ⊂ Q × Q and is invariant under the product G × G-action on Q × Q. A smooth function A : U → G is called a discrete connection form if A(q 0 , q 0 ) = e for all q 0 ∈ Q and it satisfies for all (q 0 , q 1 ) ∈ U, g 0 , g 1 ∈ G.
Using the new notion and taking Lemma 3.2 into account, we rewrite Theorem 3.4 as follows.
Theorem 3.6. Let A d be a discrete connection on the principal G-bundle π : Q → Q/G with domain U. Then, its associated discrete connection form A d is a discrete connection form. Conversely, given a discrete connection form A : U → G on the same principal bundle π, with U as in Definition 3.5, the subset Hor := {(q 0 , q 1 ) ∈ U : A(q 0 , q 1 ) = e} ⊂ Q × Q defines a discrete connection on π whose associated discrete connection form is A.
Remark 3.7. Theorem 3.6 establishes a correspondence between discrete connections and discrete connection forms. The assignments A → Hor A := A −1 ({e}) and Hor A d → A d given by (3.1) are the corresponding opposite operations. This correspondence justifies using the name A d for both the discrete connection and the discrete connection form.
Remark 3.8. The situation described in Example 3.3 corresponds to the general discrete connection on the (trivial) principal G-bundle p 1 : R×G → R. Indeed, if A d is such a connection with domain U, using (3.4), we have that A d ((r 0 , g 0 ), (r 1 , g 1 )) = g 1 A d ((r 0 , e), (r 1 , e))g −1 0 . Define C(r 0 , r 1 ) := A d ((r 0 , e), (r 1 , e)) for all (r 0 , r 1 ) ∈ U ′′ = (π × π)(U), so that Comparison of this last expression with (3.2) shows that the discrete connection form satisfies A d = A C d . As all principal G-bundles are locally of the form p 1 : R × G → R, Example 3.3 provides a local description of arbitrary discrete connections on principal G-bundles.

Discrete horizontal lift
As in the case of connections on principal bundles, discrete connections establish local diffeomorphisms between slices of the horizontal submanifold and the base space of the bundle. The inverse operation is the discrete horizontal lift.
Theorem 4.4. Let A d be a discrete connection on the principal G-bundle π : Q → Q/G with domain U. Then the following assertions are true.
(2) h d : U ′ → Q × Q is smooth and G-equivariant for the G-actions l Q×(Q/G) and l Q×Q .
Conversely, assume that U ′ ⊂ Q × (Q/G) is an open set that satisfies condition 1 (with U ′ replaced by U ′ ) and h : U ′ → Q × Q is a map such that conditions 2, 3 and 4 are satisfied (with U ′ and h d replaced by U ′ and h). Then, there exists a unique discrete connection Proof. In general, id Q × π is G-equivariant for the actions l Q×Q and l Q×(Q/G) . When A d is a discrete connection, Hor A d is G-invariant, so that (id Q × π)| HorA d is a G-equivariant diffeomorphism and, consequently, its image U ′ is G-invariant and its inverse h d is smooth and G-equivariant. This proves points 1 and 2. Point 3 follows immediately from the definition of h d , while point 4 is a consequence of ∆ Q ⊂ Hor A d . Now assume that U ′ and h are as in the statement. Let U : , that is open due to the openness of U ′ and the smoothness of id Q × π. Furthermore, it follows from the G-invariance of U ′ that U is G×G-invariant and, by the first part of condition 4, it contains the diagonal ∆ Q ⊂ Q × Q. Motivated by (4.2), define A d : U → G by A d (q 0 , q 1 ) := κ(p 2 (h(q 0 , π(q 1 ))), q 1 ). Being a composition of smooth functions, A d is smooth. Straightforward computations show that A d satisfies condition (3.4) and A d (q 0 , q 0 ) = e for all q 0 ∈ Q. All together, by Theorem 3.4, A d defines a discrete connection with domain U on π : Q → Q/G. It is immediate that U ′ = (id Q × π)(U) = U ′ and, using (4.2) as well as the definition of A d , we conclude that h = h d , proving the last part of the Theorem. The uniqueness assertion follows from the fact that the discrete connection form is determined by the horizontal lift using (4.2).
Motivated by the previous analysis, it is convenient to introduce the following notion.
Definition 4.5. Let G be a Lie group acting on Q by l Q in such a way that π : Q → Q/G is a principal G-bundle and let U ′ ⊂ Q × (Q/G) be an open subset. A smooth function h : U ′ → Q × Q is a discrete horizontal lift on π if the following conditions hold.
We can rewrite Theorem 4.4 using the new concept as follows.
Theorem 4.6. Let A d be a discrete connection on the principal G-bundle π : Q → Q/G with domain U. Then h d : U ′ → Q × Q as defined by (4.1) is a discrete horizontal lift on π. Conversely, if h : U ′ → Q × Q is a discrete horizontal lift on π : Q → Q/G, there exists a unique discrete connection A d with domain U = (id Q × π) −1 (U ′ ) on π such that U ′ = U ′ and h = h d .
Remark 4.7. We see from Theorem 4.6 that mapping A d → h d and h → A with A(q 0 , q 1 ) := κ(p 2 (h(q 0 , π(q 1 ))), q 1 ) establishes a bijection between discrete horizontal lifts and discrete connections forms -or discrete connections-on a principal bundle.
Remark 4.8. Given a discrete connection A d on the principal bundle π : Q → Q/G, its discrete connection form A d and discrete horizontal lift h d may not be defined everywhere. Indeed, if h d : Q × (Q/G) → Q × Q (that is, U ′ = Q × (Q/G)), then for any q ∈ Q, the map r → h q d (r) is a global section of the principal bundle π, so that the bundle is trivial. Hence, for nontrivial principal G-bundles h d and, consequently, A d can only be defined in some open set of the total space.
Remark 4.9. The equations of motion of a G-symmetric mechanical system on Q can be written using a connection A on the principal G-bundle π : Q → Q/G. In [1], Cendra, Marsden and Ratiu use the same type of connection to construct an isomorphic model for the reduced space T Q/G. Given a connection A, they define α A : T Q/G → T (Q/G) ⊕ g, an isomorphism of vector bundles over Q/G, by where g is the adjoint vector bundle of g. This identification allows them to establish a reduced variational principle and the associated reduced equations of motion.
Similarly, a discrete connection A d on the principal bundle π : Q → Q/G can be used to construct an isomorphic model for the discrete reduced space (Q × Q)/G.
whereG := (Q × G)/G is the adjoint bundle of Q by G with respect to the G-action on Q × G given by l Q×G g (q, h) := (l Q g (q), ghg −1 ). This identification of spaces is used in [4], [3], and [10] to study the reduction of discrete mechanical systems with symmetries. It is important to observe that, according to Remark 4.8, this identification is only local for nontrivial G-bundles.

Existence of discrete connections
All principal G-bundles carry connections that can be constructed, for instance, using Riemannian metrics on the bundle. In this section we prove an existence result for discrete connections given appropriate Riemannian metrics.
Let π : Q → Q/G be a principal G-bundle and , Q a G-invariant Riemannian metric on Q. The vertical bundle V has an orthogonal complement, the horizontal bundle H ⊂ T Q; it is easy to check that H defines a connection A , Q on the principal G-bundle π. Then there is a unique metric , Q/G on Q/G that turns π into a Riemannian submersion, i.e., for each q ∈ Q, dπ(q)| Hq : H q → T π(q) (Q/G) is an isometry.
By Theorems 8.7 in Chapter III and 3.6 in Chapter IV in [7], for every r ∈ Q/G, there is an open set W r ⊂ Q/G such that any two points in W r can be joined by a unique length minimizing geodesic (with respect to , Q/G ) lying in W r . Furthermore, for each r ′ ∈ W r there is a normal coordinate neighborhood centered at r ′ containing W r . Define that is open because the sets W r are open and π is continuous. Also, from the definition, U is G×G-invariant (for the product G-action) and contains the diagonal ∆ Q .
Let (q 0 , q 1 ) ∈ U. As π(q 0 ), π(q 1 ) ∈ W r for some r ∈ Q/G, they can be joined by a unique geodesic γ : [0, 1] → W r such that γ(0) = π(q 0 ) and γ(1) = π(q 1 ). Being A , Q a connection on the principal G-bundle π : Q → Q/G, by Proposition 3.1 in [7], there is an A , Q -horizontal lift 1 γ : [0, 1] → Q| Wr of γ(t) to Q such that γ(0) = q 0 and γ(1) ∈ Q π(q1) . In addition, as γ is a geodesic and π is a Riemannian submersion, by Proposition 3.1 in [6], γ is a geodesic for , Q . The value γ(1) is independent of the open set W r chosen for the construction. Indeed, if we pick another open set W r ′ containing π(q 0 ) and π(q 1 ) we would have two length minimizing geodesics γ r and γ r ′ joining π(q 0 ) to π(q 1 ) and contained in W r and W r ′ respectively. Then, by Theorem 10.4 in [12], γ r ([0, 1]) = γ r ′ ([0, 1]), so that both geodesics are contained in W r ∩ W r ′ and, by the uniqueness of the geodesics in W r , γ r (t) = γ r ′ (t) for all t ∈ [0, 1]. Choosing a family {W r : r ∈ Q/G} as above and constructing U with (5.1), Theorem 5.2. Let (Q, , Q ) be a Riemannian manifold where the Lie group G acts by isometries in such a way that π : Q → Q/G is a principal G-bundle. Then, there is a discrete connection A , Q d on π whose domain is U = U and whose discrete connection form is given by (5.2).
Proof. From the previous construction, U ⊂ Q × Q is an open set, G × G-invariant for the product G-action and contains the diagonal ∆ Q . In addition, by the smooth dependence of the geodesics on both the initial and final point as well as the initial point and velocity, A , Q d is smooth. For any q 0 ∈ Q, the unique length minimizing geodesic joining π(q 0 ) to itself in Q/G is the constant path, so that its horizontal lift is, again, the constant path and we conclude that Let (q 0 , q 1 ) ∈ U and g 0 , g 1 ∈ G. for any r ∈ Q/G such that π(q 0 ), π(q 1 ) ∈ W r , there is a unique length minimizing geodesic γ : [0, 1] → Q/G contained in W r and such that γ(0) = π(q 0 ) and γ(1) = π(q 1 ). As π(q 0 ) = π(l Q g0 (q 0 )) we can consider the horizontal lifts γ q0 and γ l Q g 0 (q0) of γ starting at q 0 and l Q g0 (q 0 ) respectively. As G acts on Q by isometries, for any g 0 ∈ G, l Q g0 is an isometry of Q. Then, l Q g0 ( γ q0 (t)) is a horizontal geodesic in Q starting at l Q g0 (q 0 ). Furthermore, π(l Q g0 ( γ q0 (t))) = π( γ q0 (t)) = γ(t), so that l Q g0 ( γ q0 (t)) is the horizontal lift of γ starting at l Q g0 (q 0 ), that is, γ l Q g 0 (q0) = l Q g0 • γ q0 . Then is the canonical projection, D ⊂ T Q is an open subset -containing the image of the zero section and contained in the domain of the exponential mapping-and H is the horizontal distribution.
Example 5.4. Let H be the R-algebra of quaternions together with its canonical inner product , H and basis {1,î,,k}. The submanifold of unit norm quaternions is the sphere S 3 with the round metric, while the unit norm imaginary quaternions form S 2 with the round metric. Define φ : S 3 → S 2 by φ(q) := qîq, where q denotes the conjugated quaternion (change the sign of the imaginary part of q). It is a well known fact that φ is a principal U (1)-bundle for the U (1)-action on S 3 given by the isometries l S 3 e iθ (q) := (cos(θ)1 + sin(θ)î)q. This bundle is the Hopf bundle. The map φ is a Riemannian submersion if we consider twice the round metric in S 2 .
Remark 5.5. In the same general setting as in Theorem 5.2, Example 4.1 of [9] constructs a function A d : Q × Q → G as follows. Given two points q 0 , q 1 ∈ Q, let q 01 be the geodesic in Q satisfying q 01 (0) = q 0 and q 01 (1) = 1 (this may actually restrict the domain of A d to an open neighborhood of the diagonal in Q × Q). Then, let x 01 := π • q 01 and q 01 the horizontal lift (with respect to the horizontal distribution H) of x 01 to Q starting at q 0 . Finally, define (5.3) A d (q 0 , q 1 ) := κ( q 01 (1), q 1 ).
Pairs in the "horizontal manifold" A d −1 ({e}) are the endpoints of horizontal geodesics in Q. It follows that this manifold is essentially the same as Hor Evaluation of the first derivatives of those expressions at θ = 0 confirms that (3.4) does not hold in this case.