Constrained Rough Paths

We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d-dimensional manifold and rough paths on d-dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eeels and Elworthy and Malliavin.


Introduction
In the series of papers [23][24][25], Lyons introduced and began the development of the theory of rough paths on a Banach space W. This theory allows us to model the evolution of interacting systems, driven by highly irregular non-differentiable inputs, modelled as differential equations driven by a rough path X. The theory of rough paths provides existence and uniqueness of solutions to such equations; moreover, the solutions depend continuously on the driver X. Among the many applications arising from the interplay of rough paths and stochastic analysis are the study of solutions to stochastic differential equations driven by Gaussian signals; see, for example, [2-4, 6, 15] and the analysis of broad classes of stochastic partial differential equations (PDEs) [1,8,19,20]. Rough paths also provide us with alternative ways to think about and encode the information presented in a dynamical system.
A rough path of order p ∈ [2, 3) on [0, T ] with values in a Banach space (W, | · |) is a pair of functions X s,t := (x s,t , X s,t ) ∈ W ⊕ W ⊗ W, which may be thought of as the increments of the path itself and a second-order term X s,t . Rough paths are characterised by an algebraic property analogous to the homomorphism property of the Chen series of a path (also known as the multiplicative property) and an analytic p-variation type constraint on X. A great variety of stochastic classical processes may be lifted to rough paths. For example, every R d -valued continuous semi-martingale (x s ) 0 s T (for example, Brownian motion) and large classes of Gaussian processes including fractional Brownian motion (fBM) with Hurst parameter H > 1/4 may almost surely be augmented by a process X s,t . For continuous semi-martingales, for example, one may define X s,t := where the integral can either be interpreted as an Itô integral or a Fisk-Stratonovich integral. The resulting X s,t typically does depend on which integral is used.
In view of the fact that many (if not most) natural dynamical systems come with geometric constraints, it is natural and necessary to develop a theory of constrained rough paths, that is, rough paths on a manifold, M. There is some literature in this direction (see, for example, [17,26,27]) in which rough paths theory is used to define certain stochastic evolutions on a manifold. Nevertheless, these papers sidestep the issue of actually defining the notion of a rough path in a manifold. The analogue of the approach of these papers in the smooth category would be to proceed as follows. First, consider M d as an embedded submanifold in R N for some N and then only consider curves σ : [0, 1] → M that satisfy a differential equation in the ambient space (R N ) of the formσ where z : [0, 1] → R D is a smooth curve, and {V i } D i=1 are smooth vector fields on R N such that V i (x) ∈ T x M for all x ∈ M. As a consequence, only after one has figured out how to present σ in the form of equation (1.1) is one allowed to talk about integration along a 1-form, parallel translation, unrolling, etc. For example, if α is a 1-form on M, then the approach above would define the path integral of a 1-form α along σ by Clearly, this is unsatisfactory. For example, we certainly would like to know the integral is independent of the chosen presentation of σ in equation (1.1). Moreover, from a practical and computational point of view, carrying around all of this extra structure would at best be very cumbersome and at worst would be an obstruction to developing analysis on paths and loops into a manifold, which are so prevalent in the context of Riemannian geometry and geometric PDEs. The paper [5] is the first (and until now the only) paper which develops a consistent theory of rough paths on a manifold by viewing them as a sort of non-linear current space. The theory developed in [5] has the advantage of being global and intrinsic, but it does require a rather stringent condition that the manifold be 'Lip-γ'. In this paper, we will remove this assumption and at the same time provide a concrete realisation of the currents appearing in [5]. Moreover, it is our goal to carefully develop the tools needed to make the calculus of rough paths on manifolds effective and practical for future applications.
In this paper, we suppose that M d is a d-dimensional smooth manifold which has been embedded into some Euclidean space E = R N . Our first order of business is to 'identify' those weakly geometric rough paths X ={(x s , X st ) : 0 s t T } ⊂ E ⊕ E ⊗ E which are 'constrained' to lie in M. This point is a bit subtle as the 'obvious' definition that x s ∈ M and X st ∈ T xs M ⊗ T xs M is not the correct notion. Indeed, the condition that X st ∈ T xs M ⊗ T xs M is too strict and will essentially only hold for constant rough paths. The key starting point of this paper is Definition 3.15, which basically states that a weakly geometric rough path X =(x s , X st ) ∈ E ⊕ E ⊗ E is constrained to M if and only if 1-forms on M can consistently be integrated along X. It is then shown in Section 3 that this condition is equivalent to X st being 'approximately' in T xs M ⊗ T xs M ; see Corollary 3.32. A number of other equivalent characterisations of Definition 3.15 are also given in Section 3.
Our definition of a rough path is natural in that it is the maximal class that permits a consistent definition of rough integration and, though our proofs and definitions will sometimes depend on the embedding, we will however show that the choice of embedding is not important; see Corollaries 3.40, 3.41, and Definition 3.42. In fact, the theory is intrinsic to the manifolds; in a forthcoming paper we clarify the relations of intrinsic and embedded definitions of rough paths on manifolds.
In the flat finite-dimensional setting it is known that the notions of weakly geometric and geometric rough paths are essentially equivalent. On the other hand, in the manifold setting presented here, the analogous result is not a priori known, and must be proved, which is not done until Theorem 4.17. After this point it would be possible to replace some of our proofs with smooth approximation arguments. However, we choose to avoid doing so as these approximation arguments obscure the interesting second-order differential geometric identities which underlie the theory. We believe this is important as the smooth approximation arguments will not be available when, in the future, one goes to transfer these results to infinite-dimensional settings.
The paper is organised as follows. Section 2 is devoted to introducing fundamental definitions and some preliminary results in Banach space-valued rough path theory. Section 3, as described above, is where we define a rough path in an embedded manifold. Section 4 is devoted to the notion rough differential equations (RDEs) on M. Theorem 4.2 shows that one may solve a RDE on M by extending the vector fields defining the differential equation to the ambient space, and then applying the Euclidean rough path theory to the resulting dynamical system. The output is a weakly geometric rough path in M which does not depend on any of the choices made in the extensions. Later in Theorem 4.5, we derive an equivalent intrinsic characterisation of these solutions. Here π * (U) denotes the pushforward of U by π; see Proposition 3.38.
In Section 6, we show in Corollary 6.12 that there is (similar to the smooth theory) a oneto-one correspondence between rough paths on the orthogonal frame bundle O(M ) to M and rough paths on the Euclidean space R d × so(d). Furthermore, Theorem 6.18, and Corollary 6. 19 show there are one-to-one correspondence between rough paths on M, 'horizontal' rough paths on O(M ) (see Definition 6.14), and rough paths on R d . These results are rough path versions of the stochastic rolling construction of Brownian motion on a manifold via the orthogonal frame bundle as first appeared in Eells and Elworthy [10] and then further developed in Elworthy [11][12][13] and in Malliavin [30]. We expect that the results of this paper will help lay the foundation of future work that explores the properties of manifold-valued solutions to stochastic differential equations driven by Gaussian processes such as fBMs.
The paper is completed with two appendices. In Appendix A, we gather together some needed results of the Banach space-valued rough path theory, while Appendix B explains a few details on how to view O(M ) as an embedded submanifold which are needed in Section 6 of the paper.

Basic notations
In this section, we introduce some basic notations for rough paths on Banach spaces. In addition, we gather some elementary preliminary results that will prove useful in the sequel. Some additional rough path theory results on Banach spaces needed in this paper may also be found in Appendix A. Throughout this section, V , W, and U will denote real Banach spaces. For simplicity in this paper, we will typically assume that all Banach spaces are finitedimensional. If (V, | · |) is a Banach space, then we will abuse notation and write | · | for one of the tenor norms on V ⊗ V. Because dim V < ∞, the choice of tensor norm on V ⊗ V is unimportant. For X ∈ V ⊗ V, we denote its symmetric and anti-symmetric part to be X s and X a , respectively. The following definition and (abuse of) notation will frequently be used in the sequel.
which we make into an algebra by using the multiplication in the full tensor algebra and then disregarding any terms that appear in V ⊗3 ⊕ V ⊗4 . . . . In more detail, if a, b ∈ R, x, y ∈ V, and X, Y ∈ V ⊗ V, then (a, x, X)(b, y, Y) := (ab, ay + bx, aY + x ⊗ y + bX).
In the future, we will typically write a + x + X for (a, x, X).
is a bilinear form with values in a vector space W, then, by the universal property of the tensor product, there is a unique linear map,B : Given A ∈ V ⊗ V, it will be useful to abuse notation and abbreviateB(A) as Throughout this paper, we let T denote a positive finite real number, p be a fixed real number in the interval [2,3) and ω be a control whose definition we now recall. (1) the Chen identity; that is, where here, as throughout, x s,t := x t − x s will denote the increment of the path x over [s, t]; (2) a p-variation regularity constraint: In the following, we will refer to the path x also as the trace of the rough path. We can identify a rough path as a map taking values in the tensor algebra.
Remark 2.5. It is often convenient to identify a rough path, X =(x, X), with the function Using this identification, Chen's identity becomes the following multiplicative property of X : where multiplication is given as in Definition 2.1.
The collection of V -valued p-rough paths controlled by ω is denoted by R p ([0, T ], V, ω) (also denoted by R p (V ) where no confusion arises).
Example 2.6. Suppose x : [0, T ] → V is a continuous bounded variation path. Then a simple example of a p-rough path is the (truncated) signature (S 2 (x) s,t := X s,t | 0 s < t T ) defined by and the latter integral being the Lebesgue-Stieltjes integral. For the control, we may take In this case, X is not an extra piece of information but is in fact determined by the basic path x.
Remark 2.7. If x : [0, T ] → V is continuous and of bounded variation and X is given as in equation (2.5), then as a consequence of the fundamental theorem of calculus the symmetric part of X st satisfies In this paper, we are interested in the following two important subsets of R p (V ).
(1) We say that X is a geometric p-rough path, and write X ∈ G p (V ) if X belongs to the closure of the set {Y : Y =S 2 (y), y continuous and of finite 1-variation} with respect to the topology induced by the metric (A.1).
(2) We say that X is a weakly geometric p-rough path, and write X ∈ W G p (V ) if equation (2.6) holds. [16,Corollary 8.24]) and so one typically does not have to pay much attention to the difference between geometric and weakly geometric rough paths. However, in infinite dimensions the compactness argument used in the proof that W G p (V ) ⊂ G q (V ) breaks down.

Approximate rough paths and integration
The following notation will be used heavily in this paper. Notation 2.10 ( and δ ). Let ω be a control, and assume that g and h are continuous functions from Δ [0,T ] into some Banach space W . Then we will write g s,t h s,t if there exists δ > 0 and a constant C(δ) > 0 such that, for all s and t in [0, T ] satisfying |s − t| δ, then we have If we wish to emphasize the dependence on δ, then we will write g s,t δ h s,t . Remark 2.11. As a typical application of this notation, let us note that if g : [0, T ] → V is continuous and such that g s,t 0, then, because the increments form an additive function on Δ [0,T ] , it must be that g is constant.
This elementary remark may be strengthened to apply to rough paths. The difficulty of course that the second (and higher) order processes are no longer additive with respect to (s, t). The following lemma is due to Lyons [25], and is used to powerful effect in his Extension Theorem.
Lemma 2.12. Suppose (x, X), (y, Y) ∈ R p (V ) satisfy a s,t := x s,t − y s,t 0 and A s,t := X s,t − Y s,t 0. Then the two rough paths coincide, that is, (x, X) = (y, Y). In particular, this taking (y, Y) to be the zero rough path in R p (V ), we may conclude that if (x, X) ∈ R p (V ) satisfies x s,t 0 and X s,t 0, then x st = 0 and X s,t = 0 for all 0 s t T.
Proof. Since a s,t is additive, we must have, for every partition D of [s, t], and hence x s,t = y s,t . It follows from [29,Lemma 3.4] that A s,t is also additive and repeating the argument with A s,t in place of a s,t yields the claim.
We say a functional Z :=(z, Z) defined by is an almost rough path if it satisfies the requirements of Definition 2.4 except identity (2.1), but instead holds, that is, it approximately satisfies the multiplicative identity (2.1). The following theorem due to Lyons is a cornerstone for the development of the integration for rough paths. It states that, for every almost rough path, there exists a unique rough path that is 'close.' Note that the uniqueness follows from Lemma 2.12.
Theorem 2.13. Let Z :=(z, Z) be an almost rough path on V . Then there exists a unique rough path X =(x, X) ∈ R p (V ) such that x s,t z s,t and Z s,t X s,t .
The following result due to Lyons [25] allows us to define the integral of a rough path against a sufficiently regular 1-form.
Theorem 2.14. Suppose that Z ∈ W G p (V ) and α ∈ C 2 (V, End(V, W )) is a 1-form on V with values in W. Then there is a unique X ∈ W G p (W ) such that x 0 = 0,

7)
and In the future, we will denote this X by α(dZ) and use it as the definition for the rough integral. The proof is a consequence of Theorem 2.13. The rough path integral has a number of important properties, in particular the map taking is continuous in the rough path metric (A.1).

Rough differential equations
The following definition of a RDE is in the spirit of Davie [7] and may be found, for example, in Friz and Hairer [14,Proposition 8.4].
Definition 2.15 (RDE). Let Z ∈ W G p (W ) and Y : V → Hom(W, V ) be a C 1 -map. Then X ∈ W G p (E) solves the RDE dX = Y (x)dZ (2.9) if and only if where Existence and uniqueness of solutions for RDEs defined by sufficiently regular vector fields is due to Lyons [25]. The following theorem is an easy consequence of [16,Theorem 10.14].
The following corollary is a localisation of Theorem 2.16 which will prove useful later.
Recall that if u(s, t) := Z p−var; [s,t] for (s, t) ∈ Δ [0,T ] , then u p (s, t) is a control and, in particular, u(s, t) is continuous on Δ [0,T ] and vanishes on the diagonal. Therefore, if ε := dist(U 1 , U c 2 ) > 0, then there exists (by the uniform continuity of u) a δ > 0 such that (2.13) By the choice of δ, the bound in equation (2.11), and the triangle inequality, it follows that As Y =Ỹ on U 2 , it follows that X also solves (2.12) on The solutions of rough differential satisfy a universal limit theorem which states that the map taking X to the solution Z is continuous in the p-variation metric on rough paths (see [25]). We also remark that the original definition of the solution of a rough differentiable equations (see Lyons [25]) is given in terms of a fixed point of a rough integral on V ⊕ W.
Lemma 2.18 implies that, for sufficiently regular vector fields, an RDE solution blows up if and only if both the trace and the second-order process of the solution explode. In other words, it is not possible for the explosion of a solution of an RDE to be caused only by the explosion of the second-order process of the solution.
where x 0 is given. Suppose that we can solve this equation for the trace part, that is, we can find a path x such that Proof. We can augment the trace solution x to a full rough path solution X := (x, X) as follows. Let Note that Y is bounded on x and therefore A has finite p-variation in the sense of (2.2). It now suffices to check that A is an almost multiplicative functional in the language of Lyons. For this, it will be enough to check that A approximately (in the sense of Notation 2.10) satisfies Chen's identity, which we now do. If 0 s t u T, then (2.16) Similarly, we have which is to say that A s,t is an almost multiplicative functional. Thus by Theorem 2.13 there exists X s,t such that X s,t A s,t and X s,t = 1 + x s,t + X s,t solves the RDE in equation (2.14).

Geometric and weakly geometric rough paths on manifolds
In this section, we will introduce the notions of geometric and weakly geometric rough paths on manifolds. The section is split in four parts. Subsection 3.1 introduces the basic geometric notations and facts needed for the rest of the paper.
Throughout the rest of this paper, M d will be a d-dimensional embedded submanifold of a Euclidean space E := R N . The reader may find the necessary geometric background in any number of places including [9,21,31]. To fix notation let us recall a formulation an embedded submanifold which will be most useful for our purposes.
Remark 3.4. We make a number of comments.
(3) For m ∈ U ∩ M, we have that P F (m) (Q F (m)) is the orthogonal projection onto τ m M ([τ m M ] ⊥ ) and hence is independent of the choice of local defining function. We will simply write P (m) and Q(m) (or, sometimes, P m and Q m ) for P F (m) and Q F (m) when m ∈ M.
Remark 3.5. In the proofs that follow we will often use the following identities: The last geometric notions we need are vector fields, 1-forms and their covariant derivatives.
Here we can describe the smoothness assumption of α by requiring M ∈ m → α m P m ∈ Hom(E, W ) to be a smooth function. Let Ω 1 (M, W ) denote the set of smooth 1-forms on M with values in W.
The next lemma and proposition records some basic well-known properties of the Levi-Civita covariant derivative.
Lemma 3.11. If P and Q be the orthogonal projection operators as in Remark 3.4, then dP = −dQ and P dQ = dQP.
Proof. Differentiate the identities I = P + Q and 0 = P Q, which hold on M giving the new identities in the statement.
Then the following conditions are satisfied.
which proves item 1. ( (3) If α m =α m | TmM as in item 3, then using the standard product rule again,

(Weakly) geometric rough paths on M
In the following, let M be a manifold embedded in E := R N and F be the (local) defining function as introduced in Notation 3.3. In the setting of embedded manifolds there is a natural notion of geometric rough paths that is induced by the rough metric on the ambient Euclidean space E. To help prepare the precise definition of a geometric rough path on a manifold, we introduce the following set of paths.
with respect to the topology induced by the p-variation rough path metric on E.  Proof. By definition X can be approximated by a sequence of smooth rough paths X n (see Remark 3.13) with trace in M. The traces of the approximating sequence converges in p-variation and therefore also converges pointwise. Since M is assumed to be closed, the proof is complete. Definition 3.15 (Geometric rough paths). We define geometric p-rough paths on M to be those elements ofḠ p (M ) whose trace x lies inside M. The set of geometric p-rough paths on M will be denoted by G p (M ). In other words, we have It follows from Lemma 3.14 thatḠ p (M ) = G p (M ) when M is a closed subset of E. The next example explains why it is important that we take the closure of paths in M , and why it will not be sufficient to only assume that the trace of limiting object lies in M.
Then, for any v, w ∈ R N , there exists (see [14,29]) a so-called pure area geometric rough path, X =(x, X), with the property that x = 0, the constant path zero, and An approximate version of this requirement will appear again in the general manifold setting as well; see Corollary 3.20.
A second set of rough paths on a manifold is, in structure, related to the weakly geometric rough paths in the classical Banach space setting.
Definition 3.17 (Weakly geometric rough paths). We say that X =(x, X) is a weakly geometric p-rough path on the manifold M if: X is in W G p (E), its trace x lies in M and, for any finite-dimensional subspace W and anyα ∈ Ω 1 (E, W ) such thatα| T M ≡ 0, we have α(dX) ≡ 0. The set of weakly geometric rough paths will be denoted by W G p (M ).
In the following, we will often make use of the following simple consequence of Taylor's theorem.
Proof. By Taylor's theorem, 3 ). Since f is constant on M and x, y ∈ M, it follows that f (y) − f (x) = 0 and the results follow from the previously displayed equations.
An obvious class of 1-forms having the property that α| T M ≡ 0 are those which locally have the form α = ϕF , where ϕ is a smooth function and F is a local defining function for the manifold. The following lemma gives simplified description of the level one component for the integral of any such 1-form.

is a smooth function which locally defines M as in Definition 3.1 and which has been chosen so that there is a subinterval
Proof. The product rule (written in the notation introduced in equation (3.6)) gives This identity combined with equation (2.7) then implies Since F is symmetric and X = (x, X) is a weakly geometric rough path, it follows that ,v ] and, therefore, by Lemma 3.18, Combining this estimate with equation (3.8) gives (3.5). Lemma 3.19. Then, for s u v t, where Q is defined in Notation 3.3 and Remark 3.4.
Then α| T M = 0 and therefore α(dX) ≡ 0. By Theorem 2.14, Lemma 3.19 and the fact that for s u v t and the left member of equation (3.9) is proved. This also easily proves the left member of equation (3.10) since The other approximate identities in equations (3.9) and (3.10) follow similarly; one need only now define Remark 3.21. The conditions in equations (3.9) and (3.10) are equivalent. Indeed, the proof of Corollary 3.20 has already shown equation (3.9) implies equation (3.10). For the converse direction, we need only observe that F (x u ) = F (x u )Q(x u ) so that, for example, Proof. The result follows by observing that X s,t = (P xs + Q xs ) ⊗ (P xs + Q xs )X s,t and then using equation (3.10) The following lemma prepares the definition of the integral of a rough path against smooth 1-forms. (3.11) Proof. The 1-form, ψ := α − β ∈ Ω 1 (U, W ), vanishes on T M and so by Definition 3.17, ψ(dX) ≡ 0. As the rough path integral is linear on Ω 1 (U, W ) at level one, it immediately follows that Moreover, by Corollary 3.22, The last two displayed equations along with Lemma 2.12 now gives equation (3.11).
Another proof of this lemma could be given along the lines of Proposition 3.29. The previous lemma justifies the following definition of integration α ∈ Ω 1 (M, W ) along a weakly geometric rough path X ∈W G p (M ).
Definition 3.24. The rough path integral of a rough path X ∈W G p (M ) along a smooth 1-form α ∈ Ω 1 (M, W ) is defined by Y = α(dX) as in Theorem 2.14, (3.12) whereα ∈ Ω 1 (U, W ) is any extension of α to a 1-form on some open neighbourhood M in E.
[Later, in Proposition 3.29, we will show how to characterise α(dX) without using any extension of α.] As a corollary, we immediately see that the rough integrals against smooth 1-forms are sufficient to characterise a rough path.
So for thisα it follows that A s,t =α xs (A s,t ) 0 and the result follows from Lemma 2.12.
Analogous to the Banach space setting every geometric p -rough path on a manifold is a weakly geometric p -rough path.
Proof. Let X ∈G p (M ). For the first claim, we note by definition there exists a sequence x n of smooth paths in M such that the lifts X n : as n → ∞. By definition the trace x lies in M, and it is immediate that we have X ∈W G p (M ). For the second claim, we approximate X in p variation to deduce that (3.13) holds provided α| T M ≡ 0.
In the following, we will frequently rely on localisation arguments.
is an open cover of x([0, T ]). Furthermore, since x is uniformly continuous, we can find δ = δ(X) > 0, such that, for all s and t in the interval [0, T ] with |s − t| < δ, the path segment for some i ∈ {1, . . . , k}.
The next result describes the constraints on x s,t which arise when X ∈ W G p (M ), also see Example 3.30.
Solving equation (3.16) for Q xs x st completes the proof after using the identity, wherein the last inequality made use of Lemma 3.11 and the fact that P 2 = P. It is easily seen that this agrees with (3.15).
We conclude this section with a theorem that provides a more explicit description of the integral of 1-forms along X ∈ W G p ([0, T ], M) which require no extensions of the 1-form to the ambient space. and where ∇α is the Levi-Civita covariant derivative of α as in Definition 3.10.
Proof. By Definition 3.24, Let us now use Remark 3.2 to locally extend P to a neighbourhood of M so that P = P • π. By replacingα byαP if necessary, we may assumeα =αP. Under this assumption, equation From item 4 of Proposition 3.12,

Characterising weakly geometric rough paths on M
The goal of this subsection is to show for all 0 t T and that either of the equivalent equations (3.9) or (3.10) holds locally. This will be carried out in Proposition 3.35. The next example shows that the result in Lemma 3.28 is really about paths in x t ∈ M and not so much about its augmentation to a rough path. Remark 3.5) to this equation then shows which allows us to rewrite equation (3.23) as So if x t ∈ M for all t, the component of x s,t orthogonal to τ xs M is determined modulo terms of order |x s,t | 3 by knowing the component of x s,t tangential to M at x s .
Proof. Note that by the definition of it is sufficient to check (3.25) locally for all 0 < s < t < T such that |t − s| < δ and some δ > 0. Let {U i : i = 1, . . . , k} and F i as in Remark 3.27 be a cover of the trace x. By construction of the cover for all 0 s < t T with |s − t| < δ there exists U i such that (3.14) holds. By (3.1), we may assume that, Applying item 1 of Lemma 3.18 to the right member of this equation gives the estimate Corollary 3.32. If X is an element of W G p (E) such that the trace x is in M, then the following are equivalent: [u, v], whenever F is a local defining function for M on U in the sense of Definition 3.1, and the path segment of x over [u, v] satisfies Proof. The equivalence of items 1 and 2 is an immediate corollary of Lemma 3.31. The equivalence of items 2 and 3 is the content of Remark 3.21.
Remark 3.33. If X ∈W G p (E), then the condition (I E ⊗ Q(x s ))[X s,t ] 0 is equivalent to the condition that (Q(x s ) ⊗ I E )[X s,t ] 0, that is, it does not matter in which slot the projection acts. To see this is the case, we let F : where in the second to last line we have used The next proposition shows Definition 3.17 and Definition 3.34 for the notion of a weakly geometric rough path are equivalent.
Definition 3.34 (Projection definition of weakly geometric rough paths). We say that X =(x, X) is a weakly geometric p-rough path on the manifold M if X is in W G p (E), its trace x lies in M and X satisfies wherein Q is the orthogonal projection onto the normal bundle as in Notation 3.3 and I E is the identity map on E. [ We have to show, for any finite-dimensional vector space W, that The proof will proceed in several stages, considering first 1-forms with specific structures, and finally combining those results to deduce the general claim. In what follows, we let we learn that where, for the last approximation, we have used the assumption on equation (3.27). Similarly, Equations (3.29) and (3.30) along with Lemma 2.12 shows y s,t = 0 and Y s,t ≡ 0 for all s and t.
. We conclude by using case 1 and a suitable application of Remark 3.27.
Case 3. Now assume that β ∈ Ω 1 (E, W ) is a 1-form such that β(m) ≡ 0 for all m ∈ M. If σ(t) is a path in M, then β(σ(t)) = 0 and therefore 0 = (d/dt)β(σ(t)) = β (σ(t))σ(t). Since σ(t) ∈ M is arbitrary, it follows that (∂ vm β) = 0 for all v m ∈ T m M. Hence we conclude that With this in hand, using Lemma 3.18 and equation (3.27) again, we find that (3.31) As usual, this together with the additivity of the trace shows [ β(dX)] 1 s,t = 0. Then, working as above, the second-order process is given by We further have, using An application of Lemma 2.12 then shows Y s,t ≡ 0.
The defining property in equation (3.26) is local and we therefore need a remark analogous to Lemma A.1, which allows us to concatenate rough paths on manifolds.
Remark 3.36 (Gluing). Suppose that D = {0 = t 0 < t 1 < · · · < t n = T } is any partition of [0, T ]. Let δ > 0, and suppose that the overlapping intervals J k for 1 k n are defined by Assume, for each k, we are given X(k) ∈ W G p (J k , M) such that X(k) s,t = X(j) s,t for s, t ∈ J k ∩ J j and any k and j. Then, fixing a starting point x 0 ∈ M, there exists a unique X ∈ W G p ([0, T ], M) with x(0) = x 0 which is consistent with the X(k)s in the sense that for all 1 k n,

Pushforwards and independence of the choice of embedding
Analogous to the Banach space setting (see Subsection A.2) we may consider the pushforward of rough paths on manifolds under sufficiently smooth maps.
Proof. We take each item in turn.
(1) If ϕ : M → N is a smooth map between embedded submanifolds, then it may be viewed (at least locally) as the restriction of a smooth map from Φ : E → E . It then follows that dΦ is an extension of dϕ to a neighbourhood of M and therefore, by Definition 3.24, ϕ * (X) = Φ * (X), and hence from Lemma A.4 we have that which vanishes on T M. We deduce from Definition 3.17 and 1. that ϕ * (X) ∈ W G p (N ).
Example 3.39. Suppose that ϕ : M → M is the identity map; then ϕ = Φ| E , where Φ : E → E is the identity map and therefore, The preceding example is a special case of the more general fact that diffeomorphisms give rise to bijections between the respective sets of weakly geometric rough paths on two embedded manifolds. The following corollary is immediate from Proposition 3.38.   . An alternative, more explicit, proof of the independence of the embedding for the rough paths will be given in a forthcoming paper by Cass, Driver, and Litterer where another intrinsic notion of rough paths will be developed.

RDEs on manifolds and consequences
In this section, we consider RDEs constrained to M ; see Note that Y a (x s ) ∈ T xs M and therefore there exists a smooth curve σ(t) ∈ M such thaṫ σ(0) = Y a (x s ) and we then compute This comment shows that the above definition makes sense but it is not yet clear that there  Recall that X solves equation (4.4) if and only if Using the first approximate identity in equation (4.6) along with F Ỹ = 0 shows Since F (x s ) is symmetric and Z is a geometric rough path, it follows that equation (4.5) and Combining equations (4.5), (4.7), and (4. and therefore X ∈ W G p ([0, τ), M) and we have proved local existence to equation (4.4). This shows local existence to equation (4.1). Suppose that we have found X ∈ W G p ([0, τ), M) solving equation (4.1) on [0, τ) for some τ T. If there exists a compact subset K ⊂ M such that {x(t) : t < τ} ⊂ K, then there exists t n ∈ [0, τ) such that t n ↑ τ and x ∞ := lim n→∞ x(t n ) exists in K ⊂ M. We now let U be a precompact neighbourhood of x ∞ ∈ M and F : U → R k be a local defining function of M as in Definition 3.1 and, as above, letỸ a := P F [Y a • π] on U. Moreover, we may assume thatỸ is compactly supported. By Corollary 2.17, there exists an ε > 0 and a neighbourhood V ⊂ U of x ∞ such that, for any s ∈ [τ − ε, τ ] and y ∈ V, there existsX ∈W G p ([s, τ + ε], E) with trace in U solving We then choose n sufficiently large so that t n ∈ [τ − ε, τ ] and letX ∈W G p ([t n , τ + ε], E) solve the previous equation with y = x(t n ). We may now apply the concatenation Lemma A.2 to glue X andX together to show that there exists a solution to equation (4.1) on [0, τ + ε].
Let us now consider the case where equation We now prepare an equivalent intrinsic characterisations of an RDE solution. The following proposition is a consequence of the universality property of the full tensor algebra,   (1) X solves the RDE in equation (4.1); (2) for any finite-dimensional vector space W and any α ∈ Ω 1 (M, W ),

9)
and and Combining this approximate identity with the product rule for covariant derivatives in item 2 of Proposition 3.12 gives equation ( wherein we have used P xs Y (·) (x s ) = Y (·)a (x s ) in the last equality.
(2 =⇒ 3) Applying item 2 with α = df shows This shows that item 3 holds once we recall that (3 =⇒ 1) Let W = E and f : M → E be the restrictions of the identity map on E, that is, and so equation (4.11) becomes which is precisely equation (4.2). Similarly, equation (4.12) becomes which is equivalent to equation (4.3) by Corollary 3.22.
Remark 4.6. If we restrict W to be R in Theorem 4.5, then we may still conclude from either of items 2 or 3 of that theorem that X satisfies equation (4.2), that is, the level one condition for the RDE solution (4.1).
As this true for all ∈ E * we may conclude equation (4.2) holds.
from which we may only conclude that This is because α ⊗ α(ξ ⊗ η − η ⊗ ξ) = α(ξ)α(η) − α(η)α(ξ) = 0 since scalar multiplication is commutative. Here we have used that R ⊗ R R ∼ = R. The reader should further observe that information contained in equation (4.13) is already a consequence of equation (4.2) and the assumption that Z and X are weakly geometric rough paths.

Definition 4.8 (Intrinsic RDEs on manifolds)
. Given a linear map Y : R n → Γ(T M), we say that a geometric rough path X ∈W G p (M ) solves the RDE if and only if equations (4.11) and (4.12)hold for all f ∈ C ∞ (M, W ) and every finite-dimensional vector space W.
Notation 4.9 (Intrinsic RDEs). To emphasize when we are working with the intrinsic definition of an RDE, we sometimes write in place of (4.14) where now Z s,t = z s,t + Z s,t and we interpret We end this subsection with a result describing (in special cases) the pushforward of solutions to RDEs.
Theorem 4.11. Suppose π : M → N is a smooth map between manifolds. Let Y M : R n → Γ(T M) and Y N : R n → Γ(T N) be two π-related dynamical systems. Further, suppose that Z ∈W G p (R n ) and X = (x, X) solves the RDE Proof. Fix a finite-dimensional vector space W and let f ∈ C ∞ (N, W ). Applying item 3. of Theorem 4.5 to the function f • π ∈ C ∞ (M, W ) shows

Fundamental properties of rough paths on manifolds
Armed with well-defined notions of integration and RDEs, we now derive some of the fundamental properties of geometric and weakly geometric rough paths on manifolds. We also exhibit some natural examples of elements in W G p (M ) which are constructed by 'projecting the increments' of geometric rough paths on E to the tangent space of M.
Example 4.12 (Projection construction of geometric rough paths). Let Z be a weakly geometric p-rough path on E for some p ∈ [2, 3); then there exists a unique rough path solution X (possibly only up to an explosion time) to the RDE Moreover, it will follow from Theorems 4.2 and 4.17 that X ∈ G p (M ) ∩ W G p (M ) for all p > p.
The following proposition shows that, in fact, all weakly geometric rough paths on M may be constructed by this method. Proof. The proof amounts to showing that X = Z solves equation (4.15), that is, that wherein we have used Lemma 3.28 for the second approximate equality above.
We now address the relation between geometric and weakly geometric rough paths on manifolds. To do this we first require a couple of elementary lemmas.
Proof. By replacing V by V ∩ U, we may assume that V ⊂ U.  and, moreover, such that S 2 (x k ) converges to X in W G p (M ). Consequently X ∈G p (M ).

Proof. By
By replacing V by V ∩ U, we may assume that V ⊂ U. We can then find a linear map R n a →Ỹ a ∈ Γ(T U), such thatỸ a =Ŷ a on V and the vector fieldsỸ a have compact support. As x([0, T ]) ⊂ V and X solves dX = Y dZ (x), it follows that X also solves dX =Ỹ dZ (x). By Lemma 4.14, we know the equationsẋ In addition, it follows by the universal limit theorem [29,Theorem 5.3] that solutions to the differential equations satisfy S 2 (x k ) → X in p-variation as k → ∞ and hence x k → x uniformly. Therefore, for sufficiently large k, it follows that x k (t) ∈ V for all 0 t T and hence x k (t) ∈ M (Lemma 4.14).
Since Y a =Ỹ a on V ∩ M, we conclude that x k solve (4.19), as required. Proof. We have already demonstrated the first containment in Corollary 3.32. Suppose now Z ∈ W G p ; then in particular Z ∈ W G p (E) and hence by classical results (see [16,Corollary 8.24]) Z belongs to G p (E). By Proposition 4.13, Z solves the RDE, (4.20) Consequently, by Lemma 4.16, Z ∈ G p (M ).
We conclude the section with the following theorem summarises three equivalent characterisations of weakly geometric rough paths on manifolds. We reemphasise that W G p (M ) are precisely those rough paths in W G p (E) that consistently integrate finite-dimensional vector space-valued 1-forms α ∈ Ω 1 (M, W ).

Right invariant RDE's on Lie groups
To illustrate some of the results above, we are going to consider RDEs on a Lie group G relative to right invariant vector fields. We assume, as is always possible, that G is embedded in some Euclidean space R N . Although we will be using the results above we will not need to know any information about the embedding other than it exists. Definition 4.19. To each Lie group G with Lie algebra g := Lie(G), let Y G : g → Γ(T G) be the linear map defined by that is, Y G ξ is the right invariant vector field on G such that Y G ξ (e) = −ξ. For u ∈ G, let R u : G → G be the diffeomorphism of G given by R u x = xu for all x ∈ G. By its very definition, we have R u * Y u ξ = Y u ξ • R u and so, by an application of Theorem 4.11, it follows that K =(k, K) := (R u ) * (H) solves dK = Y G dA (k) with k t0 = u on [t 0 , min(t 0 + , T )]. Choose t 0 ∈ (max{0, τ − ε/2}, τ) and apply the above result with u = g t0 in order to produce a weakly geometric rough path, K =(k, K), on [t 0 , min(τ + /2, T )] solving dK = Y G dA (k) with k t0 = g t0 . An application of Lemma A.2 (easily adapted to RDE on manifolds) shows that G restricted to [0, t 0 ] and K on [t 0 , min(τ + /2, T )] may be concatenated into a weakly geometric rough pathG which solves equation   homomorphism and for A = (a, A) ∈ W G p (g, ω) let

Theorem 4.20 (Global solutions to right invariant RDEs). To each
(4.23) is the unique global solution to the RDE (4.22) and H = ρ * (G), then Moreover, if G T s,t := P G (g s ) ⊗ P G (g s )G s,t and H T s,t := P H (h s ) ⊗ P H (h s )H s,t denote the tangential components of G and H, respectively, then H may also be characterised by A simple computation then shows ρ * Y G ξ = W ξ • ρ and therefore by Theorem 4.11, H ∈ W G p (H) satisfies the RDE (4.26) Using W as,t = Y H dρ(as,t) and

Parallel translation
In Subsection 5.1, we recall the definition of parallel translation along smooth curves in M along with some of its basic properties. In order to transfer these results to the rough path setting it is useful to introduce the orthogonal frame bundle (O(M )) over M which is done in

Smooth parallel translation
wherein the last equality follows by differentiating the identity, P (x(t))v(t) = v(t), and using If v(t) solves equation (5.1) with v(0) ∈ T x(0) M, then a simple calculation using equation (5.1) and Lemma 3.11 shows )v(t) = 0 by the uniqueness theorem of linear ordinary differential equations. Moreover, using P dQP = 0 (Lemma 3.11), Notation 5.2. Given two inner product spaces, V and W , let Iso(V, W ) denote the collection of isometries from V to W.
Lemma 5.9. If u(t) is the horizontal lift of a smooth path x(·) in M starting at (x(0), g 0 ), then u(t) is the unique solution to the ordinary differential equatioṅ (5.13) where V z (m) = P (m)z for all z ∈ E and m ∈ M as in Example 3.7.
Proof. A path u(t) = (x(t), g(t)) ∈ O(M ) solves equation (5.13) if and only if while the horizontal vector field associated to a ∈ R d (determined by ∇) is defined by The connection 1-form on O(M ) determined by the covariant derivative ∇ is given by where u(t) = (σ(t), g(t)) is any smooth curve in O(M ) such thatu(0) = (ξ, h) (m,g) .
we may express ω ∇ more simply as

Rough parallel translation on O(M )
As in Proposition 3.12, we may choose to write Γ for dQ. The following definition is motivated by Lemma 5.9. where V z (x) := P x z as in Example 3.7 and V ∇ z is its horizontal lift as in Definition 5.8. [In Proposition 5.15, it will be shown that equation (5.19) has global solutions, that is, U exists on [0, T ].] Lemma 5.14. If U is parallel translation along X as in Definition 5.13, then π * (U) = X.
Proof. From Definition 5.8, we know that V ∇ and V are π-related dynamical systems and therefore, by Theorem 4.11,X := π * (U) solves the RDE On the other hand, by the consistence Proposition 4.13 we know that X satisfies the same RDE and so by uniqueness of solutions to RDEs we conclude that X =X = π * (U). In particular, the RDE in equation (5.19) exists for all time that X is defined.
Proof. Using dX = V dX (x) along with item 2. of Theorem 4.5 implies E) be the projection map, f (x, g) = g. From Theorem 4.5, Combining this equation with the identities, From the theory of linear RDE [29] or by a minor modification of the results in Theorem 4.20 we know that G solving equation (5.20) does not explode. Therefore, we may then conclude that u t = (x t , g t ) has no explosion. Combining this result with Lemma 2.18 then shows that the RDE of equation (5.19) also does not explode.
then U is a parallel translation along X, that is, X = π * (U) and U satisfies equation (5.19).
Proof. Since Y ∇ and Y are π-related, it follows from Theorem 4.11 that X = π * (U). Using Theorem 4.5 and Remark 4.6, equation (5.22) at the first level is equivalent to while U solving equation (5.19) at the first level is equivalent to (5.24) where in each case F is assumed to be an arbitrary smooth function on O(M ). Thus to complete the proof we must show that equation (5.23) implies equation (5.24) and show the second-order condition Putting this together with equation (5.23) shows, Zs,t . Thus to complete the proof, we must show But we already know that X solves equation (5.21), which applied to the identity function I on R N shows which is precisely equation (5.26).
We will actually be more interested in the following variant of Theorem 5.16.
Then U is a parallel translation along X = π * (U), that is, U satisfies equation (5.19).
Proof. Working as above, equation (5.27) is equivalent to while U solving equation (5.19) is equivalent to equation (5.24), where in each case F is assumed to be an arbitrary smooth function on O(M ). Thus to complete the proof, we must show that equation (5.28) implies equation (5.24) and the correspondence of the second-order pieces by the approximate identity First recall that and putting this together with equation (5.28) shows Applying equation (5.28) to F = π, where π(x, g) = x, shows

Rolling and unrolling
In this section, we develop the rough path analogy of Cartan's rolling map. As a consequence we will see that rough paths on a d-dimensional manifold are in one-to-one correspondence with rough paths on d-dimensional Euclidean space.
is a linear isomorphism for all m ∈ M. We refer to any choice of Y : R d → Γ(T M) with this property as a parallelism of M. Associated to a parallelism Y is an R d -valued 1-form on M given by It is easy to see that every vector space is parallelizable; we detail some other not so trivial examples which will be useful later. Example 6.2. Every Lie group G is parallelizable. Indeed, if we let d = dim G, so that the Lie algebra g := Lie(G) ∼ = R d , then Y = Y G of equation (4.21) defines a parallelism on G. In this case, the associated 1-form θ Y is known as the (right) Maurer-Cartan form on G.
In this case, we can define a parallelism by taking where B a and V A were defined in equations (5.14) and (5.15). In this case, the associated where θ and ω are as in Definition 5.11.

Smooth rolling and unrolling
The following 'rolling and unrolling' theorems in the smooth category are all relatively easy to prove and therefore most proofs are omitted here. They are included as a warm-up to the more difficult rough path versions which appear in the next subsection.
Conversely, given z ∈ C 1 0 ([0, T ], R d ), the solution to the differential equatioṅ The solution to (6.4) determines the inverse of the map (6.3); that is, the solution to (6.4) satisfies (6.5) and any x ∈ C 1 o ([0, τ), M) agrees with the solution w to the differential equatioṅ until the explosion time of this equation.
and define u to be the solution to the differential equationu which may explode in finite time τ := τ (a, A) < T. Then u is in C 1 uo ([0, T ], O(M )), and over [0, τ) we have · 0 (θ, ω)(du) = (a · , A · ). (6.8) Theorem 6.6. The solution to (6.7) determines the inverse to (6.6) until explosion; that is, the solution to (6.7) satisfies (6.8), and any u ∈ C 1 uo ([0, T ], O(M )) agrees with w, the solution to the differential equatioṅ  The solution to (6.12) determines the inverse to (6.11) until explosion; that is, both (6.13) holds and any X ∈ W G p (M, o) agrees with W the solution to the RDE

Rough rolling and unrolling
until the explosion time of this equation.
Proof. Suppose that Z ∈ W G p (R d , 0) and let X solve equation (6.12).
= 0 and hence from item 2 of Theorem 4.5 Conversely, suppose that X ∈W G p (M, o) and now define Z =(z, Z) by Z = · 0 θ Y (dX). We need to show, making the usual caveat about explosion, that X is the solution to (6.14). To this end, we first note Z s,t [θ xs ⊗ θ xs ][P xs ⊗ P xs ]X s,t and z s,t θ xs P xs x s,t + ∇θ[P xs ⊗ P xs ]X s,t .
Since Y θ = Id T M , it follows from the last two equations that or, equivalently, that and Again, using the fact that Y θ = Id T M , we see that which combined with equation (6.16) and the fact that θY a = a for all a ∈ R d implies This gives equation (6.18) since Q(∂ Y P )Y = Q∂ Y Y, which is proved by applying Q to the identity

Rolling via the frame bundle
We can specialise this result to O(M ). Making use of the notation in Example 6.3, we obtain the following.
until the explosion time of this equation.
Definition 6.14. We say a rough path , (0, 0)), (6.22) where ω is the connection 1-form defined in (5.17)   Proof. Recall Γ = dQ and that U solves (see Definition 5.13) dU = V ∇ dX (u), where V ∇ a (m, g) = (V a (m), −Γ(V a (m))g) and V a (m) = P m a for all a ∈ E. Using these formulas, we find, for u = (m, g) ∈ O(M ) and a, b ∈ E, that wherein in the last line we have used P g = g so that g * = g * P and hence g * Γ(V a (m))V b (m) = g * P m dQ(V a (m))P m V b (m) = 0.
From these identities and item 2 of Theorem 4.5, we conclude from which it follows that , (0, 0)). where θ is the canonical 1-form.

Appendix A. Some additional rough path results
In this section, we gather some additional results and notation of the theory of rough paths on Banach spaces. The literature on Banach space-valued rough paths is now so well-established as to be classical; the reader seeking more background has a great many choices: [14,16,18,22,28,29]. As in Section 2, let V, W, and U denote Banach spaces. In addition, we assume p ∈ [2, 3) is a fixed number and ω is a control in the sense of Definition 2.3.
Recall the definition of a p-rough path and R p (V ), the set of p-rough paths on V from Definition 2.4. We can define a metric on R p (V ) by setting for X =(x, X), Y =(y, Y) ∈ R p (V ). Note that endowed with this metric R p (V ) is a complete metric space.

A.1. Concatenation of local rough paths on M
Localisation plays an important role in the manifold setting, and we need results which will allow us to glue together locally constructed rough paths on M. The following elementary lemma (compare [5]) allows us to concatenate a finite number of rough paths.
Lemma A.1 (Concatenating rough paths). Suppose that Π = {0 = t 0 < t 1 < · · · < t n = T } is a partition of [0, T ]. For k ∈ {1, . . . , n}, let J k := [t k−1 , t k ], and, for each k, assume that we are given X(k) ∈ W G p (J k , W ). Then there exists a unique X ∈ W G p ([0, T ], W ) such that x(0) = 0 and for all 1 k n, X(k) s,t = X s,t for all s, t ∈ J k . (A.2) Proof. Let x(0) = 0. For 0 s t T with s ∈ J k and t ∈ J , we define X s,t := X(k) s,t k X(k + 1) t k ,t k+1 · · · X( ) t −1 ,t , where we now view X(k) u,v ∈ 1 ⊕ W ⊕ W ⊗ W and the multiplication is the usual multiplication in the truncated tensor algebra (see, for example, [29]). We now need to check that X ∈ W G p ([0, T ], W ). The multiplicative property of rough paths follows directly from equation (A.3). The weakly geometric property can either be verified by direct calculation or one just observes that a rough path is weakly geometric if and only if it has finite p-variation and takes values in the free nilpotent group of step p (see, for example, [29, p. 53]). We finally check that X satisfies the correct variation conditions. To this end, observe that if ω is a control so that |x u,v | = |x u,v (k)| ω(u, v) 1/p and |X 2 u,v | ω(u, v) 2/p for u, v ∈ J k , 1 k n, a straightforward calculation shows that there exists a constant C p,n such that C p,n ω(s, t) controls the concatenated path.
The following lemma allows us to compose the flows of RDEs. Proof. We only have to check the definition of an RDE solution for time s < τ < t, that is, times s < t which straddle τ. We write X = (x, X) for the concatenated path and G(x) for Y (x)Y (x). We have x s,t = x s,τ + x τ,t Y (x s )z s,τ + G(x s )Z s,τ + Y (x τ )z τ,t + G(x τ )Z τ,t Y (x s )z s,τ + [Y (x s ) + Y (x s )x s,τ ]z τ,t + G(x s )Z s,τ + G(x s )Z τ,t Y (x s )[z s,τ + z τ,t ] + [Y (x s )Y (x s )z τ,t ]z τ,t + G(x s )Z s,τ + G(x s )Z τ,t = Y (x s )z s,t + G(x s )[z τt ⊗ z τ,t + Z s,τ + Z τ,t ] = Y (x s )z s,t + G(x s )Z s,t (Chen's identity).
The second-order term is simpler; we have X s,t = X s,τ + X τ,t + x s,τ ⊗ x τ,t as desired.

A.2. Pushforwards of rough paths
In this subsection introduce the notion of a pushforward of a rough path between two Banach spaces and record its elementary properties (cf. also [5]). Definition A.3. Suppose that ϕ ∈ C 2 (W, V ) and Z ∈ W G p (W ); then the push-forward of Z by ϕ is defined by ϕ * Z := ϕ(z 0 ) + dϕ(dZ).
In more detail we are letting Note that ϕ * Z ∈W G p (V ). The first level of the pushforward of a rough path has a more explicit representation.
As both ends of this equation are continuous additive functionals, we may conclude using Remark 2.11 that [ϕ * Z] 1 s,t = ϕ(z t ) − ϕ(z s ).
Theorem A.5 (Integration of pushforwards). Suppose that Z ∈ W G p (W ), ϕ ∈ C 2 (W, V ) and α ∈ C 2 (V, End(V, U )) is a 1-form on V with values in U. Then Proof. By definition β := ϕ * α is a U -valued one form on W which is determined by β(z)v = α(ϕ(z))ϕ (z)v ∈ U for all z, v ∈ W.