SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA: TIME-SYMMETRY WITHOUT SPACE-SYMMETRY

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1 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA: TIME-SYMMETRY WITHOUT SPACE-SYMMETRY TOM DUCHAMP, GANG XIE, AND THOMAS YU Abstract. This paper estabishes smoothness resuts for a cass of noninear subdivision schemes, known as the singe basepoint manifod-vaued subdivision schemes, which shows up in the construction of waveetike transform for manifod-vaued data. This cass incudes the (singe basepoint) Log-Exp subdivision scheme as a specia case. In these schemes, the exponentia map is repaced by a so-caed retraction map f from the tangent bunde of a manifod to the manifod. It is known that any choice of retraction map yieds a C scheme, provided the underying inear scheme is C (this is caed C equivaence ). But when the underying inear scheme is C 3, Navayazdani and Yu have shown that to guarantee C 3 equivaence, a certain tensor P f associated to f must vanish. They aso show that P f vanishes when the underying manifod is a symmetric space and f is the exponentia map. Their anaysis is based on certain C k proximity conditions which are known to be sufficient for C k equivaence. In the present paper, a geometric interpretation of the tensor P f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponentia map. The condition P f = 0 is shown to be equivaent to the condition that f agrees with the exponentia map of the connection up to the 3rd order. In particuar, when f is the exponentia map of a connection, one recovers the origina connection and P f vanishes. It then foows that the condition P f = 0 is satisfied by a wider cass of manifods than was previousy known. Under the additiona assumption that the subdivision rue satisfies a time-symmetry, it is shown that the vanishing of P f impies that the C 4 proximitiy conditions hod, thus guaranteeing C 4 equivaence. Finay, the anaysis in the paper shows that for k 5, the C k proximity condtions impy vanishing curvature. This suggests that vanishing curvature of the connection associated to f is ikey to be a necessary condition for C k equivaence for k Introduction Motivated by the vast deveopment in waveet anaysis and the proiferation of manifod-vaued data in severa areas of science and engineering, such as diffusion tensor imaging and motion capturing, Donoho et a [11, 0] introduced a framework for a noninear waveet transform for mutiscae representations of data iving on a manifod, which he assumed was either a Lie group or a symmetric space. Underying this framework is a noninear subdivision scheme on a manifod M of the form ( ) (1.1) (Sx) h+σ = exp xh a +σ og xh (x h ), σ = 0, 1, h Z, Date: August 4, 011 revised: Apri 7, Mathematics Subject Cassification. 41A5, 6B05, E05, 68U05. Key words and phrases. Noninear subdivision, Affine connection, Retraction, Exponentia map, Riemannian manifod, Symmetric space, Curvature, Time-symmetry. Tom Duchamp gratefuy acknowedges the support and hospitaity provided by the IMA during his visit from Apri to June 011, when much of the work in this artice was competed. Gang Xie s research was supported by the Fundamenta Research Funds for the Centra Universities and the Nationa Natura Science Foundation of China (No ). Thomas Yu s research was partiay supported by the Nationa Science Foundation grants DMS and DMS He is aso indebted to a feowship offered by the Louis and Bessie Stein famiy. 1

2 TOM DUCHAMP, GANG XIE, AND THOMAS YU where (a ) is the mask for a inear subdivision scheme S in, exp : T M M is the exponentia map of M, og x is the oca inverse of exp x : T M x M, the restriction of exp to T M x, and x = {x h } is a sequence of points 1 in M. (Reca that exp x is a diffeomorphism between a neighborhood of 0 x T M x and a neighborhood of x M.) Notice that the noninear scheme S depends on three data: the underying manifod M, the map exp, and the inear subdivision rue S in. A basic probem in anaysis is to determine conditions under which S shares the same smoothness as the underying inear scheme S in. This is the so-caed smoothness equivaence probem. In previous work [31, 9, 5, 8], it was found that Donoho s origina conjecture that S is aways as smooth as S in is most ikey not true in genera. The conjecture does hod in the foowing two cases: (i) If S in (and hence aso S) is interpoatory, then S and S in are C k equivaent for any k [8, 5, 14]. (ii) If we use two different (and carefuy constructed) basepoints x i and x i+1/ for the even and odd rues, then a modified version of (1.) satisfies the C k equivaence property for arbitrary k [9, 15]. Neither the interpoatory nor the two basepoint scheme is desirabe for the waveet-ike transform in [0]: The former eads to L -instabiity aready in the inear setting (see the unpubished artice [9]), whie the atter forces us to give up non-redundancy (a.k.a. critica samping in the waveet iterature.) For these reasons, we consider here the singe basepoint pane scheme (1.1)) with S in non-interpoatory. In this paper we prove the foowing: (iii) The non-interpoatory singe basepoint scheme S can satisfy C k equivaence up to C 4, but our anaysis indicates that, for many manifods of interest, the equivaence is doomed to breakdown at degree 5. One woud expect the interpoatory and the non-interpoatory schemes to have simiar smoothness equivaence properties. Aso, it is surprising that the smoothness equivaence properties of the singe basepoint strategy are so different from those of the two basepoint strategy in the atter strategy, the choice of retraction (see beow), time-symmetry, and curvature pay no roe in the anaysis, but as we sha see, a three pay a roe in smoothness properties of the singe basepoint subdivision scheme. Our anaysis is based on a generaization of (1.1). Let M denote any smooth, n-dimensiona manifod without boundary. Let 0 x denote the zero tangent vector based at x M. Reca that exp(0 x ) = x for a x M and that its restriction exp x : T M x M is a oca diffeomorphism between a neighborhood of 0 x in T M x and a neighborhood of x in M. We repace the exponentia map by a smooth map f : T M M, satisfying these two conditions. Thus, the restriction f x : T M x M has a oca inverse g(x, ) : V x T M x with g(x, x) = 0 x, where V x M is an open neighborhood of x; and we can now define a noninear subdivision rue as foows: ( ) (1.) (Sx) h+σ = f xh a +σ g(x h, x h ), σ = 0, 1, h Z. We sha view M as a subset of T M by identifying each point x M with 0 x T M, the zero tangent vector based at x. With this identification, the restriction of f to M is the identity map, and so f is a smooth retraction in the standard topoogica sense. Because in this paper we ony consider retraction maps whose restrictions to T M x are oca diffeomorophisms, we abuse notation and refer to this specia cass of maps maps as retraction maps. (This is consistent with the terminoogy in the appied mathematics iterature [3, ].) In Section we give a more detaied discussion of such maps. 1 For Sx to be we defined, adjacent points must be sufficienty cose. We impicity assume this condition throughout.

3 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 3 Reca that a subdivision scheme is caed C k if it yieds C k curves for any initia data. We say that the noninear scheme S has the C k equivaence property if it is C k whenever the underying inear scheme S in is C k Main resuts. In [31] it was shown that S has the C 3 equivaence property if the retraction map f satisfies a condition of the form P f = 0, where P f is a certain tensor constructed from f and independent of S in ; and numerica evidence was presented suggesting that C 3 equivaence fais for P f 0, and it was conjectured that the condition P f = 0 is necessary and sufficient for C 3 equivaence. In this paper, we show that the condition P f = 0 has a simpe geometric interpretation. As we sha see in Section, a retraction f defines a torsion-free affine connection, which in turn defines an exponentia map, which we denote by exp f. In the case where the retraction is the exponentia map of a torsion free, affine connection, we recover the origina connection; but in genera f and exp f ony agree to second order aong M T M, as iustrated in the foowing diagram: (1.3) f. (affine connection). exp f.. agree up to nd order aong M T M In Section 3, we present a simpified proof that the condition P f = 0 is sufficient for S to have the C 3 equivaence property. We then prove that P f = 0 if and ony if f agrees with exp f to the 3rd order aong M T M. This enabes us to show that the condition is satisfied not ony by the (standard) exponentia maps defined on Lie groups and symmetric spaces, but aso by the exponentia map of any torsion-free affine connection on any manifod. This significanty generaizes the resuts of [31, Theorem 8] and shows that, the ony roe payed by the symmetric space structure is through its exponentia map, the symmetric space structure, itsef, has itte to do with the C 3 equivaence condition. In particuar, the condition P f = 0 hods true if f is the standard exponentia map on any Riemannian manifod and even more generay if f is the exponentia map of any torsion free, affine connection on any manifod. We next consider C 4 equivaence. Using the proximity conditions of [9], we prove in Section 4 that the condition P f = 0 impies C 4 equivaence provided that the underying inear subdivision scheme has a natura time-symmetry. In the absence of time-symmetry, we show that the proximity conditions force the curvature of the affine connection associated with f to vanish and aso force the retraction f to agree with exp f up to 4-th order. Athough in numerica anaysis imposing a natura symmetry in a numerica scheme often impies an additiona order of accuracy, it is however surprising that this can happen without any requirement on the 4th order behavior of f. Finay, in Section 5, we prove that the C 5 proximity conditions impy vanishing curvature, even for inear schemes with a time-symmetry. It is we-known that vanishing curvature imposes stringent conditions on the topoogy of the underying manifod. By a cassica resut of Ausander and Markus [4], if a manifod has a compete, torsion free, fat affine connection then its universa cover is R n. This means that many manifods of interest do not have a retraction that satisfies the C 5 proximity conditions. In particuar, the C 5 proximity condition is automaticay vioated on a spheres, a non-abeian Lie groups, and a Grassmannians. Moreover, even in cases, such as GL(n), where the manifod admits a torsion free, fat affine connection, the natura retraction map may be rued out. One can show, for exampe, that the exponentia maps on GL(n) and the symmetric space P OS n of positive definite symmetric matrices both define affine connections with non-vanishing curvature. We remark that the above resuts appy, in particuar, to the case where f is the exponentia map of a torsion-free affine connection. Consequenty, our resuts appy to the important specia cases where f is the exponentia map of a Lie group or a symmetric spaces. They aso appy to certain homogeneous spaces that are not symmetric spaces (see [19] for detais), and to a Riemannian manifods, and to a affine manifods...

4 4 TOM DUCHAMP, GANG XIE, AND THOMAS YU The reader shoud note, however, that the proximity conditions we study here are ony known to be sufficient conditions for C k equivaence. We conjecture that they are aso necessary conditions, but this remains an open probem. In Section 6, we discuss necessity in a specia case. 1.. Time- and Space-Symmetry. In this and our previous paper [31], we use the term time-symmetry to refer to an invariance property of a subdivision scheme under a time -reversa t t in the domain. This form of symmetry comes in two kinds: prima and dua. For any (inear or noninear) subdivision scheme S, we say that S has a prima time-symmetry if S R 0 = R 0 S where R 0 is the refection operation about 0, i.e. (R 0 x) k = x k. Simiary, we say that S has a dua time-symmetry if S R 1/ = R 1/ S where R 1/ is the refection operation about 1/, i.e. (R 1/ x) k = x 1 k. We summarize inear subdivision schemes with these two kinds of time-symmetry in Tabe 1. Prima Dua Exampes: Odd degree B-Spine, Even degree B-Spine, Dubuc s scheme Donoho s AI scheme Data: Associated with dyadic points Assocated with dyadic intervas Property of S in : S in R 0 = R 0 S in S in R 1/ = R 1/ S in Property of mask: a k = a k a 1 k = a k Property of refinabe function: φ( x) = φ(x) φ(1 x) = φ(x) Tabe 1. Prima and dua time-symmetry for inear subdivision schemes The term space-symmetry, on the other hand, refers to invariance of the subdivision scheme under a transitive group action on the range space. In the inear case, where the range space is R n, we of course have S in (Ax + b) = AS in x + b for any affine transformation x Ax + b in R n even when S in does not possess any time-symmetry. More generay, when the range space is a homogeneous space M acted upon by a transitive group action G, then the space-symmetry refers to the property S(g x) = g S(x) for a g G. Athough the proof in [31] of the main resut makes essentia use of space-symmetry and whie space-symmetry is a desirabe property in practice, we sha see in Section 3 and 4 of this paper that the main resuts in [31] are vaid without assuming any space-symmetry. On the other hand, we sha aso see in Section 4 that the notion of dua time-symmetry in the inear subdivision scheme underying the noninear S (1.) pays an interesting roe in the smoothness properties of S hence the tite of this artice. In Donoho s origina use of the Log-Exp scheme, the underying inear subdivision schemes is either an interpoatory Desauriers-Dubuc scheme, which has a prima time-symmetry, or an average-interpoating (AI) subdivision scheme [10], which has a dua time-symmetry. In the former case, we know from previous resuts [9, 5, 8] that, due to the interpoatory property, S is aways as smooth as S in. So Donoho s conjecture is true in this specia case. For the atter case, the resuts in this paper te us that, athough Donoho s origina smoothness equivaence conjecture is most ikey incorrect, C k equivaence hods in the practica range k 4.

5 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 5. Retraction maps Reca from the introduction that the noninear subdivision schemes we consider here are defined in terms of a retraction map f. In this section, we show how to construct a torsion free affine connection f associated to each retraction map. Let B T M be an open neighborhood of the set of zero vectors and et f : B M be a smooth map satisfying the condition f(0 x ) = x for a x M. Let f x : B x := T M x B M denote the restriction of f to the tangent space to M at x M. We assume that f satisfies the additiona requirement that f x : B x M is a diffeomorphism onto its image. In oca coordinates, we can express f in the form (.1) f : R n R n R n : (x, X) f(x, X) = (f 1 (x, X),..., f n (x, X)). with f(x, 0) = x for a x. Because f(x, 0) = x, the Tayor expansion of f with respect to X at X = 0 has the form (.) f (x, X) = x + f i (x) X i + 1! f ij(x) X i X j + 1 3! f ijk(x) X i X j X k +..., where fi (x) = f (x,0) X, f i ij (x) = f (x,0) X i X, etc. j Remark 1. The Tayor expansion of f in (.) is for a fixed x but varying X. If we consider the more genera Tayor expansion of f in oca coordinates with both x and X varying, then we encounter the mutiinear maps (see [31, Section ]) F α,β : R n R }{{ n } α times defined in a component-free, basis-independent way by the formua (.3) F α,β (u 1,..., u α ; v 1,..., v β ) = d ds 1 d ds α We sha use the notation F α,β extensivey in the next section. R n R }{{ n R } n β times d d ( f x + dt 1 dt β si=t j=0 α β s i u i, i=1 j=1 t j v j ). Let e i, i = 1,..., n denote the standard basis for R n. Then in oca coordinates, using the component-wise notation with Einstein convention, F 0,1 (X) = f i (x)x i e F 0, (X, Y ) = f ij(x) X i Y j e, F 0,3 (X, Y, Z) = fijk(x) X i Y j Z k e F 1, (X; Y, Z) = f jk (x) x i X i Y j Z k e for tangent vectors X = X i e i, Y = Y i e i, and Z = Z i e i. In this notation, the Tayor expansion (.) assumes the form (.4) f(x, X) = x + F 0,1 (X) + 1! F 0,(X, X) + 1 3! F 0,3(X, X, X) A standard comptuation shows that the map A f : T M T M : X F 0,1 (X) is coordinate independent. Reca that f x : B x M is a diffeomorphism from a neighborhood of 0 x to a neighborhood of x. By the Inverse Function Theorem, this impies that the n n matrices f j i (x) are a invertibe, so A f : T M T M is an automorphism of T M. We may, therefore, use A f to normaize f as foows. Let B 0 = A f (B) and et f = f A 1 f. Then f is aso a retraction map, and by construction, A f is the identity map.

6 6 TOM DUCHAMP, GANG XIE, AND THOMAS YU Moreover, f defines the same noninear subdivision rue as f does. To see this note that the oca inverse of f x is g (x, ) = A f g(x, ), and so the subdivision rue S defined by f is given by ) ( ) ( (S x) h+σ = f x h a +σ g (x h, x h ) = f A 1 f ( a +σ A f (g(x h, x h ))) ( ) = f a +σ g(x h, x h ) = S(x) h+σ, where we have used inearity of the map A f. Therefore, without oss of generaity, we may assume that A f is the identity. This eads to the foowing forma definition: Definition. A retraction is a smooth map f : B M, defined on a neighborhood of the zero vectors such that f(0 x ) = x for a x M and such that A f = id T M : T M T M. For the remainder of this paper we assume that f satisfies Definition. Consequenty, the Tayor expansion (.) reduces to the form (.5) f (x, X) = x + X + 1! f ij(x) X i X j + 1 3! f ijk(x) X i X j X k Equivaenty, using component-free notation f(x, X) = x + X + 1! F 0,(X, X) + 1 3! F 0,3(X, X, X) Remark 3. From a computationa point of view, a retraction is usuay regarded as an approximation to the standard exponentia map on a matrix Lie group or a symmetric space; from this point of view, the exponentia map comes first, and the approximating retraction comes afterward. For exampe, the foowing diagona Padé approximations R m,m of e z e z = z z 1 1 z + 1 +O(z 5 ) = z 1 }{{ z 1 1 } z +O(z 3 ) }{{} R,(z) R 1,1(z) have the remarkabe property that they map so(n) to SO(n), and, when combined with the group operation on SO(n), then can be used to define retractions on SO(n) that are cheaper to compute than the exponentia map, see [31, Section 4.4]..1. The affine connection of a retraction. We next show that the quantities Γ k ij := f ij k in Equation.5 define a torsion-free, affine connection on M. It suffices to check that the quantities fij satisfy the foowing transformation identity for connection coefficients: { (.6) Γ k ij = xk x a x b x c x i x j Γc ab + x c } x i x j, where Γ c ab are the connection coefficients in x-coordinates. To see this, et x be another set of oca coordinates, et x = φ(x) denote the change of coordinates map, and et f(x, X) denote the expression for f in x-coordinates. Taking into account the change of coordinates formua for tangent vectors, X j ( ) ( ) x j x j x j = x i Xi x j = Xi x i x j = X i x i,

7 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 7 gives the foowing identity reating f and f: ( ) (.7) f (x, X) = f φ(x), x1 x j Xj,..., xn x j Xj = φ ( f 1 (x, X),..., f n (x, X) ). Differentiate Equation.7 twice with respect to X i and X j at X = 0 and use the chain rue to obtain the formua f (x, 0) xa x b X a X b x i x j = x f a (x, 0) f b (x, 0) x a x b X i X j + x f k (x, 0) x k X i X j (Note that we have used the identity f i (x,0) X j using the fact that the Jacobian matrix ( x x k ) is invertibe, gives the fina transfor- Finay, soving for fij k mation identity, = x x i x j + x x k f k (x, 0) X i X j. f x a x b ab x i x j = x x i x j + x x k f ij k. = δ j i.) Rewriting this in terms of f k ik and f k ij gives f k ij = xk x c Setting Γ k ij = f ij k yieds precisey the identity (.6). { x a x b x i x j ( f c ab) + x c } x i x j. Because Γ k ij are mixed partia derivatives, the connection is torsion-free, i.e. Γk ij = Γk ji for a i, j, k. We summarize the above discussion in the next emma. Lemma 4. Every retraction f : B M induces a torsion-free affine connection on M, with connection coefficients given in oca coordinates x = (x 1,..., x n ) by the formua Γ k ij(x) = f k (x, 0) X i X j... The exponentia map of an affine connection. In this section, we reca some standard facts about the exponentia map of an affine connection. See [16] for a more compete exposition. Assume that Γ k ij are the connection coefficients of any affine connection (not necessariy defined by f). Consider the initia vaue probem: (.8) ẍ + Γ i,jẋ i ẋ j = 0, x(0) = x, ẋ(0) = X for X = (X 1,..., X n ) R n. The soution γ X (t) = (x 1 (t),..., x n (t)) of (.8) is caed an autoparae curve. Because γ X is the soution of a differentia equation with smooth coefficients Γ k ij, it depends smoothy on the initia condition. Aso, for X sufficienty sma, γ(t) is defined for 0 t 1. So for sufficienty sma X, the equation exp(x) = γ X (1) makes sense. This defines a map exp : R n B 0 R n on an open neighborhood of 0 R n and it is not difficut to show that it satisfies the foowing properties: (i) exp(0) = x (ii) d 0 exp = id x : R n R n, We do not assume here that the connection is the Levi-Civita connection of an underying Riemannian metric. Consequenty, these curves are not necessariy geodesics in the sense of Riemannian geometry.

8 8 TOM DUCHAMP, GANG XIE, AND THOMAS YU where d 0 exp denoted the derivative at 0 R n. We can reinterpret exp as the map exp : T M x B x M defined on a neighborhood of the zero vector 0 x T M x. A standard argument using the transformation rues for tangent vectors and connection coefficients shows that exp is coordinate independent, and etting x vary over a points on M gives a map (.9) exp : T M B M defined on an open neighborhood B of the set of zero vectors, caed the exponentia map of the connection. Remark 5. A connection is said to be compete when every autoparae curve can be extended indefinitey. It is we-known that when M is compact, a affine connections are compete. Remark 6. In coordinate-free form, properties (i) and (ii) above assume the form (i) exp(0 x ) = x for a x M (ii) d 0 exp = id : T M x T M x for a x, where the tangent space to T M x at 0 x is identified with T M x, itsef. In future sections we wi need to use a Tayor expansion for autoparae curves. Suppose that x(t) is an autoparae curve, satisfying the initia vaue probem (.8). Because x(0) = x, and ẋ(0) = X, the Tayor expansion of x(t) is x (t) = x + t X t! Γ ijx i X j +..., where we have used the differentia equation for x(t) to express ẍ(0) in terms of ẋ(0). Equation (.8) with respect to t yieds a formua for the third derivative: (.10) from which we obtain the Tayor expansion... x = Γ ij x k ẋ ẋ i ẋ j Γ ijẍ i ẋ j Γ ijẋ i ẍ j (.11) x (t) = x + tx t Γ ijx i X j t3 3! = Γ ij x k ẋk ẋ i ẋ j + Γ ijγ i stẋ s ẋ t ẋ j + Γ ijẋ i Γ j stẋ s ẋ t { } Γ ij = x k Γ isγ s jk ẋ i ẋ j ẋ k, { Γ ij x k Γ isγ s jk } X i X j X k + O(t 4 ). Differentiating Definition 7. We denote by exp f the exponentia map of the connection defined by the retraction map f. As we noted above, the exponentia map of any affine connection is, itsef, a retraction map. Consequenty, it in turn defines a torsion-free, affine connection. The next proposition shows that this process stops: Proposition 8. If exp is the exponentia map of a torsion free affine connection, then exp exp = exp. Proof. We need ony show that Γ k ij(x) = exp k (x, 0) X i X j. But Equation (.8), which defines autoparae curves, shows that exp has the Tayor expansion exp k (x, X) = x k + X k 1! Γk ij(x)x i X j +....

9 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 9 3. The Geometric Interpretation of the condition P f = 0 In [31] the foowing invariant of the retraction f was found: (3.1) P f (u) := F 0, (u, F 0, (u, u)) + 1 F 1,(u; u, u) 1 F 0,3(u, u, u), and the condition P f = 0 was shown to be sufficient for S to have the C 3 equivaence property. Here we give a geometric interpretation of this condition. As a coroary, we show that the exponentia map of every torsion-free affine connection has the C 3 equivaence property. Definition 9. We say that a retraction f : T M B M satisfies the order k condition if it agrees up to k-th order with the exponentia map exp f aong M T M. Equivaenty, f satisfies the order k condition if the Tayor expansions of the two famiies of curves on M: agree up to order k for a X B. (i) µ X : t f(tx) and (ii) γ X : t exp f (tx) In ight of the Tayor expansions (.5) and (.11), µ X (t) and γ X (t) have Tayor expansions of the forms and µ X(t) = x + tx + t! f ijx i X j + t3... µ (0) ! γx(t) = x + tx + t! f ijx i X j + t3... γ (0) +..., 3! respectivey, and so µ X (t) and γ X (t) aways agree up to second order for a X and so f and exp f aways satisfy the order condition. We now show that P f = 0 is equivaent to the condition that f satisfy the order 3 condition. Theorem 10. Let f be a retraction. Then P f = 0 if and ony if... µ X (0) =... γ X (0) for a X B. Proof. Choose an arbitrary vector X. From Equation (.11), we have { }... Γ γ ij X(0) = x k Γ isγ s jk X i X j X k, where Γ k ij = f ij k. On the other hand, from (.5), we have... µ X(0) = fijkx i X j X k. Thus,... γ X (0) =... µ X (0) if and ony if { } f (3.) fijkx i X j X k ij = x k + f isf jk s X i X j X k. On the other hand, in component-free notation, the invariant P f is given by the formua in (3.1), which in component-wise notation assumes the form Pf (x, X) = fisx i ( fjkx s j X k) + 1 fij (x) x k X i X j X k 1 f ijkx i X j X k (3.3) { } = fisf jk s + 1 fij x k 1 f ijk X i X j X k. Comparing (3.) and (3.3) shows immediatey that P f (x, X) = 0 if and ony if... γ X (0) =... µ X (0).

10 10 TOM DUCHAMP, GANG XIE, AND THOMAS YU Remarks 11. (1) Note that the terms fis f jk s and f ij are not symmetric in ijk; the summation over a x k (i, j, k) in Equation (3.3) uses ony the symmetric parts of these quantities. () Since P f (u) is a homogeneous poynomia of degree 3, by the poarization theorem (see [8, page 8] or [1]), there is a unique symmetric triinear map, which by abuse of notation we again denote by P f, such that P f (u, u, u) = P f (u): P f (u, v, w) = 1 P f (λ 1 u + λ v + λ 3 w) 3! λ 1 λ λ 3. λ=0 It is given by the formua 3 : (3.4) P f (u, v, w) = 1 3 [F 0,(u, F 0, (v, w)) + F 0, (v, F 0, (u, w)) + F 0, (w, F 0, (u, v))] + We ca P f (u, v, w) the depoarized form of P f (u). 1 6 [F 1,(u; v, w) + F 1, (v; u, w) + F 1, (w; u, v)] 1 F 0,3(u, v, w). Theorem 10, when combined with Proposition 8, has an immediate coroary: Coroary 1. Let exp : T M M be the exponentia map of a symmetric connection on M. Then the invariant P exp vanishes identicay. In particuar, if M is a Lie group, a symmetric space, or a Riemannian manifod and f : T M B M is its exponentia map, then S and S in satisfy the C 3 equivaence property. The key point of this coroary is that each of these standard exponentia maps is actuay the exponentia map of a symmetric affine connection of the underying manifod. The Riemannian case is of course weknown from the Levi-Civita connection. For the cases of Lie group and symmetric space, see Loos [17]. 4. C 4 anaysis with and without time-symmetry Given Theorem 10, it is natura to ask if the order 4 proximity condition is guaranteed by the order 4-th condition on f. This specuation turns out to be fase. In this section, we prove the foowing resut: Theorem 13. Assume that the retraction f : T M M satisfies the condition P f = 0 for C 3 equivaence. Then S and S in satisfy C 4 equivaence if either of the foowing two conditions is satisfied: (a) The inear scheme S in has a dua time-symmetry, i.e. a k = a 1 k. (b) The curvature R f of the affine connection defined by f vanishes and in addition f satisfies the order 4 condition given by Definition 9. As we mentioned in the introduction, the vanishing curvature condition rues out many manifods of interest. As such, Theorem 13 has a dichotomous favor: with time-symmetry in the inear scheme, C 4 equivaence is guaranteed without any constraint on the 4-th order behavior of f or any symmetry property whatsoever on the manifod M. Without time-symmetry, the theorem suggests that we can ony get C 4 equivaence on a fat affiney connected manifod. 3 In [31, Appendix] Pf is shown to be independent of the choice of coordinates, and therefore a triinear map of the tangent bunde of M. In differentia geometry jargon, this is aso caed a tensor fied of type (1, 3) on M.

11 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA Review of previous resuts. The proof of Theorem 13 is based on the now standard proximity approach, introduced in [4, 3]. We reca the foowing resut from [9]. Theorem 14 ([9, Theorem.4]). Assume that the inear scheme S in is stabe and C k, k 1. If S and S in satisfy the order k proximity condition 4 (in some oca coordinates), i.e. there exists a constant C > 0 such that for any dense enough bounded sequence x, we have (4.1) where (4.) Ω j (x) := then S is aso C k. j 1 Sx j 1 S in x C Ω j (x), j = 1,..., k, j γ Γ j i=1 i x γi, Γ j := { γ = (γ 1,, γ j ) γ i Z +, } j i γ i = j + 1, Remark 15. In [30], we prove that the proximity condition is invariant under change of coordinates, i.e. the proximity condition is satisfied in one coordinate system if and ony if it is satisfied in any other coordinate system. This resut wi be expoited in the proof of Theorem 13 and again in Section 5. i=1 We use Theorem 14 here in the same manner as it was used in [31] to study C 3 equivaence. Our resut is oca, so we may work in oca coordinates. We write f in the form f(x, X), with oca inverse g(x, y), satisfying g(x, x) = 0. By the ocaity and shift-invariant properties of both S and S in, it suffices to assume that the sequence x R n in (4.1) above is a finite sequence indexed by {0, 1,..., L} for a L k arge enough 5 so that x 0,..., x L determine (Sx) h+σ and (S in x) h+σ, for k consecutive indices of h + σ. Then at east one entry of the sequence k 1 Sx k 1 S in x can be determined from x 0,..., x L, which is a we need to determine. We view (Sx S in x) h+σ as an R n -vaued function of x = (x 0,..., x L ). In [31] it is shown that (Sx S in x) h+σ can be written in the form (Sx S in x) h+σ = Φ k,h+σ (D 1,..., D k 1 ) + O(Ω k (x)), where Φ k,h+σ is a certain R n -vaued poynomia in the variabes (4.3) D 1 = ( x) 0 = x 1 x 0, D = ( x) 0 = x x 1 + x 0,..., D k 1 = ( k 1 x) 0. Observe that D j = O( j x ) for j < k and D j = O( k x ) for j k. Thus, we can write k ( ) h x h x 0 = D j + O( k x ). j j=1 The poynomia Φ k,h+σ is obtained by computing the Tayor poynomia of degree k of (Sx S in x) h+σ at the constant sequence x h = x 0, changing to the variabes D 1,..., D L, and absorbing as many terms as possibe into O(Ω k (x)). To obtain a more precise expression for Φ k,h+σ, we need the foowing muti-index notation. Let q ( ) h D J := (D j1,..., D jq ), A h J :=, J := j j q. k=1 4 This is not to be confused with the order k condition in Definition 9. 5 For the smaest support C k subdivision scheme, namey the dyadic subdivision scheme coming from the degree k + 1 B-spine, L is exacty k. j k

12 1 TOM DUCHAMP, GANG XIE, AND THOMAS YU for J = (j 1,..., j q ) any ordered ist of integers between 1 and k 1. For integers α, m with 0 α m, m k, consider the set of muti-indices of the form I = (J, (n i, β i, J1, i J) i α i=1 ), where q = m α and J = (j 1, j,..., j m α ), J i 1 = (j i 1,1,..., j i 1,n i β i), J i = (j i,1,..., j i,β i), where 1 j i, j1,a, i j,a i k, 1 n i k m + 1, and 0 β i n i. We need ony consider muti-indices I satisfying the additiona condition α (4.4) I := J + ( J1 i + J ) i k. Finay, et N I = N (J,(ni,β i,j1 i,j i)α i=1 ) denote the mutiinear map ( ) (4.5) N I (D) := N (J,(ni,β i,j1 i,j i)α i=1 ) (D) = F α (m) D J ; G (n1) β 1 (D J 1 1 ; D J 1 ),..., G (nα) β α (D J α 1 ; D J α ), i=1 where D = (D 1,..., D k 1 ) are as in (4.3); and et c h,σ I [ ] [ α (4.6) c h,σ (J,(n i,β i,j1 i,j i)α i=1 ) = α A h J where F (m) α := i=1 A h J i 1 i=1 := c h,σ (J,(n i,β i,j i 1,J i )α i=1 1 (m α)!α! F m α,α (x0,0), G (n) β := a +σ A h J i be the rea number ) ] α a +σ i=1 A h J i 1 (n β)!β! G n β,β (x0,x 0), and F α,β are the derivatives of f defined in (1) and G α,β are the anaogousy defined derivatives of g. With this notation in pace, we are ready to reca the foowing emma. Lemma 16 ([31, Lemma 3]). For any k,, (4.7) (Sx S in x) h+σ = Φ k,h+σ (D 1,..., D k 1 ) + O(Ω k (x)) k = c h,σ I N I (D 1,..., D k 1 ) + O(Ω k (x)). m= I =m Two remarks are in order: Remark 17. First note that to use Lemma 16 together with Theorem 14, we need to anayze not ony Sx S in x, but aso j (Sx S in x) for a differencing orders j k 1. Assume that we have aready estabished the order k 1 proximity condition, 6 To estabish the next higher order, we need ony prove (4.1) for j = k. Now, since the spatia indices (h, σ) ony show up in (4.6), the operator k 1 ony acts on the sequences of coefficients defined by (4.6). Therefore, it is sufficient to prove that the poynomia k 1 Φ k,h+σ (D 1,..., D k 1 ) := k m= I =m k 1 c h,σ I N I (D 1,..., D k 1 ) vanishes. In ight of the k 1 proximity conditions, the sum of terms of weight ess than k aready vanishes. It, therefore, suffices to determine ony the cases of I = (J, (n i, β i, J1, i J) i α i=1 ) with I = k and for which the sequence (in h, σ) (4.8) k 1 c h,σ I is non-zero. We refer to such an index I = (J, (n i, β i, J1, i J) i α i=1 ) as a non-vanishing case. 6 Reca that S and Sin aways satisfy order proximity condition.

13 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 13 Remark 18. This emma sheds some ight on why the noninear scheme S suffers from a breakdown of smoothness equivaence. By construction, the coefficients (4.6) are poynomias in h for each fixed σ = 0 or 1. If a the sequences in (4.6) were poynomia sequences with degree not exceeding k, then the k 1-order differences in Equation (4.8) woud aways be zero. Therefore by Theorem 14, the noninear scheme (1.) woud satisfy the C k equivaence property for any k. Unfortunatey, this is too good to be true. And we may view this as a strong indication of why these schemes appear to suffer from a breakdown of smoothness equivaence (see [9, Section 1.1]). We may attribute the breakdown to the first bracket in (4.6). The probem occurs ony when at east one of the ists J and J i 1, i = 1,..., α, is non-empty. In this case, since the index σ (= 0 or 1) does not show up in the first bracket of (4.6), the sequence (4.6) is not even a poynomia sequence, but consists of two interacing poynomia sequences. In the case when a of J and J i 1, i = 1,..., α are empty, the first bracket becomes the constant unit sequence, and we are eft with the sequence in the second bracket. In this case, the sequence is a poynomia sequence (in h) and, moreover, is one which makes (4.8) vanish. This is the content of our next emma beow. Lemma 19 (Essentiay borrowed from [9]). Assume that the inear scheme S in reproduces Π k. The second square bracket on the right-hand side of (4.6) is aways a poynomia of degree not exceeding α i=1 J i. (Note aso that α i=1 J i k.) Consequenty, (4.8) vanishes when J, J i 1, i = 1,..., α are a empty; therefore we need not consider these cases. 7 We may aso ignore the cases when α i=1 J i 1, as this impies (4.6) vanishes. Proof. See Appendix A. 4.. Non-vanishing cases. In this section, we enumerate the non-vanishing cases in Lemma 16 when k = 4. Under the P f = 0 condition, we aready have the order 3 proximity condition. Therefore, we ony need to prove (4.1) for j = 4, and we ony need to consider those (J, (n i, β i, J1, i J) i α i=1 ) such that J + α i=1 J 1 i + J i = 4. Our enumeration wi be divided into two parts. Part I foows from an observation vaid for any k 4, whie Part II simpy consists of those non-trivia cases for k = 4 not incuded in Part I Part I. Fix a k 4. According to the constraints, in any non-vanishing case there are at east three D j s, each being at east 1, and (consequenty) is at most k. In fact, the ony way we can see a term invoving D k is when the map F α (m) ( (n ( ; G i) β i ( ; ) ) α i=1) arising from (4.5) is 3-inear and it acts on the arguments D 1, D 1, D k (in any order.) By the symmetries of the mutiinear maps F α (m) and G (n) β, different combinations of (J, (n i, β i, J1, i J) i α i=1 ) ead to the same term in (4.5). For exampe, since F α (m) is invariant under permutation of the ast α arguments, for any permutation σ of {1,..., α}, F α (m) ( DJ ; G (n σ(1)) β σ(1) (D σ(1) J 1 is the same. ; D J σ(1) ),..., G (n σ(α)) β σ(α) (D J σ(α) 1 ; D σ(α) J ) ) We now enumerate such terms in Tabe. The observation is that, after expoiting symmetry, there are aways 7 such terms regardess of the vaue of k. In Tabe, each of the 7 cases is assigned a case abe 7 This emma was overooked in the order 3 proximity anaysis in [31]. Notice that the zeros in the ast coumn of [31, Tabe 1] correspond exacty to the cases where a J, J1 i, i = 1,..., α are a empty.

14 14 TOM DUCHAMP, GANG XIE, AND THOMAS YU without any prime ( ); any case with a abe foowed by prime(s) is one eading to a term identica to that coming from the corresponding unprimed case. As such, each of these 7 cases has a mutipicity associated to it, which we record in the ast coumn of the tabe. Case (m, α) J (n i ) α i=1 (β i ) α i=1 (J1 i, Ji )α i=1 c h,σ F α (m) ( D J ; ( G (n i ) (D β i J i ; D 1 J i ) ) ) α i=1 Mut. (A) (, ) J (n 1, n ) (β 1, β ) J 1 1, J1, J 1, J (AI) (1, ) (AI.1) () (1, 1) (), (k ), (1), (1) Υ 1 F () (G (1) 1 (D k ), G () 1 (D 1; D 1 )) (AI.) () (1, 1) (), (1), (1), (k ) Υ 1 F () (G (1) 1 (D 1), G () 1 (D 1; D k )) (AI.3) () (1, 1) (), (1), (k ), (1) Υ F () (G (1) 1 (D 1), G () 1 (D k ; D 1 )) (AI ) (, 1) (B) (3, ) J (n 1, n ) (β 1, β ) J 1 1, J1, J 1, J (BI) (1, 1) (BI.1) (1) (1, 1) (), (1), (), (k ) Υ 1 F (3) (D 1 ; G (1) 1 (D k )) (BI.1 ) (1) (1, 1) (), (k ), (), (1) (BI.) (k ) (1, 1) (), (1), (), (1) Υ F (3) (D k ; G (1) 1 (D 1)) 1 (C) (3, 3) J (n 1, n, n 3 )(β 1, β, β 3 ) J 1 1, J1, J 1, J, J3 1, J3 (CI) (1, 1, 1) (CI.1) () (1, 1, 0) (), (k ), (), (1), (1), () Υ 1 F (3) 3 (G (1) 1 (D k ), G (1) 0 (D 1)) 6 (CI.1 ) () (1, 1, 0) (), (1), (), (k ), (1), () (CI.1 ) () (1, 0, 1) (), (k ), (1), (), (), (1) (CI.1 ) () (1, 0, 1) (), (1), (1), (), (), (k ) (CI.1 ) () (0, 1, 1) (1), (), (), (k ), (), (1) (CI.1 ) () (0, 1, 1) (1), (), (), (1), (), (k ) (CI.) () (1, 1, 0) (), (1), (), (1), (k ), () Υ F (3) 3 (G (1) 0 (D k )) 3 (CI. ) () (1, 0, 1) (), (1), (k ), (), (), (1) (CI. ) () (0, 1, 1) (k ), (), (), (1), (), (1) Tabe. The seven non-vanishing cases invoving ony D k when k 4. If k 1 Υ 1 = k 1 Υ, these cases impose a condition equivaent to the P f = 0 condition, otherwise they force us to impose the zero curvature condition. Note: When k = 3, there are ony 3 cases (see [31, Tabe 1]) which ead to the P f = 0 condition Part II. When k = 4, the possibe combinations for (m, α) in (4.7) are: (, ), (3, ), (3, 3), (4, ), (4, 3), (4, 4). In Tabe 3, we enumerate for each of these six (m, α) a the non-vanishing cases (J, (n i, β i, J1, i J) i α i=1 ) with J + α i=1 J 1 i + α i=1 J i = 4 and those not aready covered in Part I. These cases are isted in coumns 3-6 in Tabe 3. We foow a simiar convention as in Tabe. For exampe, case (AII. ) gives the same term as case (AII.). Cases (CII ) and (CII ) both ead to the same group of four terms determined by cases (CII.1)-(CII.4) Anaysis of c h,σ. It shoud be cear that (4.6) has a stronger invariance w.r.t. the subscript indices than (4.5). Whie we have a tota of different cases for (4.5) isted in Tabes -3, by inspection, there are ony two distinct sequences in (4.6) coming from Tabe : (4.9) Υ h,σ 1 := A h 1(S in A h 1S in A h k S in A h 1A h k ) h+σ Υ h,σ := A h k (S in A h 1S in A h 1 S in A h 1A h 1) h+σ

15 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 15 Case (m, α) J (n i ) α i=1 (β i ) α i=1 (J1 i, Ji )α i=1 c h,σ F α (m) ( D J ; ( G (n i ) (D β i J i ; D 1 J i ) ) ) α i=1 Mut. (A) (, ) J (n 1, n ) (β 1, β ) J 1 1, J1, J 1, J (AII) (, ) (AII.1) () (1, 1) (1), (1), (1), (1) Ξ F () (G () 1 (D 1, D 1 ), G () 1 (D 1, D 1 )) 1 (AII.) () (1, ) (1), (1), (), (1, 1) Ξ 1 F () (G () 1 (D 1; D 1 ), G () (D 1; D 1 )) (AII. ) () (, 1) (), (1, 1), (1), (1) (AIII) (1, 3) (AIII.1) () (1, 1) (), (1), (1, 1), (1) Ξ F () (G (1) 1 D 1, G (3) 1 (D 1, D 1, D 1 )) (AIII.) () (1, ) (), (1), (1), (1, 1) Ξ 1 F () (G (1) 1 (D 1), G (3) (D 1; D 1, D 1 )) (AIII ) (3, 1) (B) (3, ) J (n 1, n ) (β 1, β ) J 1 1, J1, J 1, J (BII) (1, ) (BII.1) (1) (1, 1) (), (1), (1), (1) Ξ F (3) (D 1 ; G (1) 1 (D 1), G () 1 (D 1; D 1 )) (BII.) (1) (1, ) (), (1), (), (1, 1) Ξ 1 F (3) (D 1 ; G (1) 1 (D 1), G () (D 1, D 1 )) (BII ) (, 1) (C) (3, 3) J (n 1, n, n 3 ) (β 1, β, β 3 ) J 1 1, J1, J 1, J, J3 1, J3 (CII) (1, 1, ) (CII.1) () (0, 1, 1) (1), (), (), (1), (1), (1) Ξ F (3) 3 (G (1) 0 D 1, G (1) 1 D 1, G () 1 (D 1, D 1 )) 3 (CII.1 ) () (1, 0, 1) (), (1), (1), (), (1), (1) (CII.) () (1, 1, 1) (), (1), (), (1), (1), (1) Ξ 3 F (3) 3 (G (1) 1 (D 1), G () 1 (D 1, D 1 )) 3 (CII.3) () (0, 1, ) (1), (), (), (1), (), (1, 1) Ξ 1 F (3) 3 (G (1) 0 (D 1), G (1) 1 (D 1), G () (D 1, D 1 )) 3 (CII.3 ) () (1, 0, ) (), (1), (1), (), (), (1, 1) (CII.4) () (1, 1, 0) (), (1), (), (1), (1, 1), () Ξ F (3) 3 (G (1) 1 (D 1), G () 0 (D 1, D 1 )) 3 (CII ) (1,, 1) (CII ) (, 1, 1) (D) (4, ) J (n 1, n ) (β 1, β ) J 1 1, J1, J 1, J (1, 1) (1, 1) (1, 1) (), (1), (), (1) Ξ F (4) (D 1, D 1 ; G (1) 0 (D 1), G (1) 0 (D 1)) 1 (E) (4, 3) J (n 1, n, n 3 ) (β 1, β, β 3 ) J1 1, J1, J 1, J, J3 1, J3 (E.1) (1) (1, 1, 1) (1, 1, 0) (), (1), (), (1), (1), () Ξ F (4) 3 (D 1 ; G (1) 0 (D 1)) 3 (E.1 )-(E.1 ) (E.) (1) (1, 1, 1) (), (1), (), (1), (), (1) Ξ 3 F (4) 3 (D 1 ; G (1) 1 (D 1)) 1 (F) (4, 4) J (n 1, n, n 3, n 4 )(β 1, β, β 3, β 4 )J1 1, J1, J 1, J, J3 1, J3, J4 1, J4 (F.1) () (1, 1, 1, 1) (1, 1, 0, 0) (), (1), (), (1), (1), (), (1), () Ξ F (4) 4 (G (1) 0 (D 1), G (1) 0 (D 1)) 6 (F.1 )-(F.1 ) (F.) () (1, 1, 1, 0) (), (1), (), (1), (), (1), (1), () Ξ 3 F (4) 4 (G (1) 0 (D 1)) 4 (F. )-(F. ) Tabe 3. The non-vanishing cases for k = 4 for not covered by Tabe. Note: Cases (E.1 )-(E.1 ), (F.1 )-(F.1 ) and (F. )-(F. ) refer to the obvious shuffing of (J i 1, J i ) α i=1 of their corresponding unprimed cases. and three coming from Tabe 3: (4.10) Ξ h,σ 1 := A h 1(S in A h 1S in A h 1A h 1 S in A h 1A h 1A h 1) h+σ Ξ h,σ := A h 1A h 1(S in A h 1S in A h 1 S in A h 1A h 1) h+σ Ξ h,σ 3 := A h 1(S in A h 1S in A h 1S in A h 1 S in A h 1A h 1A h 1) h+σ. Coumn 7 in both tabes indicates which of the five sequences is obtained in each case. Note that these five sequences are dependent on S in and independent of the retraction f. After appying the 3rd order differencing operator to them, there are ony 4 independent sequences; when S in has a dua time-symmetry, we are eft with ony two. Lemma 0. Let S in be a inear subdivision scheme that reproduces Π 3 and has a dua time-symmetry, i.e. a i = a 1 i, we have: (i) 3 Ξ 1 = 3 Ξ = 3 3 Ξ 3, and (ii) 3 Υ 1 = 3 Υ when k = 4 in (4.9).

16 16 TOM DUCHAMP, GANG XIE, AND THOMAS YU Proof. See Appendix B. The proof of Theorem 13 uses the foowing curvature condition. Lemma 1. The condition that the curvature of the affine connection induced by f vanishes is equivaent to the condition (4.11) F 1, (v; u, v) F 1, (u; v, v) + F 0, (u, F 0, (v, v)) F 0, (v, F 0, (u, v)) = 0, u, v. Proof. In oca coordinates using index notation, and recaing that Γ k ij = f ij k, condition (4.11) assumes the form { } Γ jk x i + Γ ik x j + Γ iaγ a j,k Γ jaγ a ik X i Y j X k x = 0 for a X, Y. We now use the foowing we-known formua for the curvature tensor of a connection (4.1) R f (X, Y )Z = R ijkx i Y j Z k x, where Rijk = Γ jk x i Γ ik x j + Γ ipγ p jk Γ pjγ p ik. which shows that (4.11) is equivaent to the condition R f (X, Y )X = 0 for a X, Y. Ceary then, vanishing curvature impies (4.11). The converse is the content of the next emma. Lemma. Suppose that R f (X, Y )Y = 0 for a X, Y. Then R f (X, Y )Z = 0 for a X, Y, Z. Proof. 8 Any biinear mapping b(u, v) with b(u, u) = 0 is skew-symmetric, so our assumption impies that the curvature R f (X, Y )Z is skew-symmetric in the variabes Y and Z. Since the curvature tensor is aso skew-symmetric in X,Y, the first Bianchi identity R f (X, Y )Z + R f (Y, Z)X + R f (Z, X)Y = 0 transforms to R f (X, Y )Z + R f (X, Y )Z + R f (X, Y )Z = 0 if we appy two swaps to the second and third terms. This shows 3R f = Proof of Theorem 13. Armed with Tabes and 3 and Lemmas 0 and 1, we are now ready to prove Theorem 13. Proof of Theorem 13. By Remark 17, we ony need to show that 3 Φ 4,h+σ vanishes identicay. First notice that the variabe D 3 does not appear in Tabe (with k = 4) nor in Tabe 3. Further inspection of the tabes shows that (4.13) 3 Φ 4,h+σ (D 1, D ) = 3 Υ 1 H Υ H + 3 Ξ 1 Q Ξ Q + 3 Ξ 3 Q 3, 8 We wish to thank one of the referees for suggesting this proof.

17 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 17 where (4.14a) (4.14b) H 1 (D 1, D ) := F () (G (1) 1 (D ), G () 1 (D 1; D 1 )) + F () (G (1) 1 (D 1), G () 1 (D 1; D )) + F (3) (D 1 ; G (1) 1 D 1, G (1) 1 D ) + F (3) 3 (G (1) 1 (D ), G (1) 0 (D 1)) 6, H (D 1, D ) := F () (G (1) 1 (D 1), G () 1 (D ; D 1 )) + F (3) (D ; G (1) 1 (D 1)) 1 + F (3) 3 (G (1) 0 (D )) 3, (4.15a) Q 1 (D 1 ) := F () (G () 1 (D 1; D 1 ), G () (D 1, D 1 )) + F () (G (1) 1 (D 1), G (3) (D 1; D 1, D 1 )) + F (3) (D 1 ; G (1) 1 D 1, G () (D 1, D 1 )) + F (3) 3 (G (1) 0 (D 1), G (1) 1 (D 1), G () (D 1, D 1 )) 3, (4.15b) (4.15c) Q (D 1 ) := F () (G () 1 (D 1; D 1 ), G () 1 (D 1; D 1 )) 1 + F () (G (1) 1 (D 1), G (3) 1 (D 1; D 1, D 1 )) + F (3) (D 1 ; G (1) 1 (D 1), G () 1 (D 1, D 1 )) + F (3) 3 (G (1) 0 (D 1), G (1) 1 (D 1), G () 1 (D 1, D 1 )) 3 + F (3) 3 (G (1) 1 (D 1), G () 0 (D 1, D 1 )) 3 + F (4) (D 1 ; D 1, G (1) 0 (D 1), G (1) 0 (D 1)) 1 + F (4) 3 (D 1 ; G (1) 0 (D 1)) 3 + F (4) 4 (G (1) 0 (D 1), G (1) 0 (D 1)) 6, Q 3 (D 1 ) := F (3) 3 (G (1) 1 (D 1), G () 1 (D 1; D 1 )) 3 + F (4) 3 (D 1 ; G (1) 1 (D 1)) 1 + F (4) 4 (G (1) 0 (D 1)) 4. These expressions simpify consideraby if we express the derivatives of g in terms of those of f, and repace D 1 and D by u and v, respectivey. By appying the chain rue to the reation f(x, g(x, y)) = y, together with our assumptions that F (1) 0 = F (1) 1 = id, we have (4.16a) G (1) 0 = id, G (1) 1 = id, G () 0 = 1 F 0,, G () = 1 F 0,, G () 1 (u, v) = F 0,(u, v), and, using (3.1), (4.16b) G (3) 1 (u; u, u) = 3 F 0,(u, F 0, (u, u)) + F 1, (u; u, u) 1 F 0,3(u, u, u) = 1 F 0,(u, F 0, (u, u)) + 1 F 1,(u; u, u), (4.16c) G (3) (u, u; u) = 3 F 0,(u, F 0, (u, u)) 1 F 1,(u; u, u) + 1 F 0,3(u, u, u) = 1 F 0,(u, F 0, (u, u)). (The biinear map G () 1 happens to be symmetric in its two arguments. The triinear maps G (3) 1 and G (3), on the other hand, are not symmetric. The asymmetries do not concern us for now as we ony need the expressions for G (3) 1 (u3 ) and G (3) (u3 ).)

18 18 TOM DUCHAMP, GANG XIE, AND THOMAS YU Now appy (4.16) to simpify the expressions for H i and Q j : (4.17a) (4.17b) H 1 (u, v) = F 0, (v, F 0, (u, u)) + F 0, (u, F 0, (u, v)) + F 1, (u; u, v) F 0,3 (v, u, u) H (u, v) = F 0, (u, F 0, (u, v)) + 1 F 1,(v; u, u) 1 F 0,3(u, u, v) (4.18a) Q 1 (u) := 1 F 0,(F 0, (u, u), F 0, (u, u)) 1 F 0,(u, F 0, (u, F 0, (u, u))) 1 F 1,(u; u, F 0, (u, u)) + 1 F 0,3(u, u, F 0, (u, u)) (4.18b) Q (u) := 1 F 0,(F 0, (u, u), F 0, (u, u)) + 1 F 0,(u, F 0, (u, F 0, (u, u))) 1 F 0,(u, F 1, (u; u, u)) + F 1, (u; u, F 0, (u, u)) 5 4 F 0,3(u, u, F 0, (u, u)) F,(u, u; u, u) 1 F 1,3(u; u, u, u) F 0,4(u, u, u, u), (4.18c) Q 3 (u) := 3 4 F 0,3(u, u, F 0, (u, u)) F 1,3(u; u, u, u) 1 4 F 0,4(u, u, u, u). Next use the condition P f (u) = 0 to further simpify Equation (4.13) as foows. Let (4.19a) P 1 (u, v) :=H 1 (u, v) + H (u, v) =F 0, (v, F 0, (u, u)) + F 0, (u, F 0, (u, v)) + F 1, (u; u, v) + 1 F 1,(v; u, u) 3 F 0,3(u, u, v), and (4.19b) P (u) :=Q 1 (u) + Q (u) + 3 Q 3(u) = 1 F 1,(u; u, F 0, (u, u)) + 1 F 0,(u, F 1, (u; u, u) 1 4 F 1,3(u; u, u, u) F,(u, u; u, u). We caim that conditions P 1 (u, v) = 0 and P (u) = 0 are satisfied and are simpy the de-poarization and the spatia differentiation of the condition P f (u) = 0. First compare (4.19a) and (3.4) to concude P 1 (u, v) = 3 P f (u, u, v) = 0. Next observe that differentiating (3.1) with respect to the spatia variabe x, gives exacty P (u), from which we concude P (u) = 0. The equations P 1 (u, v) = 0 and P (u) = 0, together impy (4.0) H 1 + H = 0, and Q 1 + Q + 3 Q 3 = 0. Consequenty, Equation (4.13) reduces to (4.1) 3 Φ 4,h+σ (u, v) = ( 3 Υ 1 3 Υ ) H 1 (u, v) + ( 3 Ξ Ξ 3 ) Q 1 (u) + ( 3 Ξ 3 3 Ξ 3 ) Q (u). To prove part (a), assume that S in has a dua time-symmetry and observe that Lemma 0 immediatey shows that 3 Φ 4,h+σ (u, v) vanishes identicay.

19 SINGLE BASEPOINT SUBDIVISION SCHEMES FOR MANIFOLD-VALUED DATA 19 To prove part (b), assume that S in does not have a dua time-symmetry and that 3 Φ 4,h+σ = 0. Setting v = 0 gives 3 Φ 4,h+σ (u, 0) = ( 3 Ξ Ξ 3 ) Q 1 (u) + ( 3 Ξ 3 3 Ξ 3 ) Q (u). It foows that ( 3 Υ 1 3 Υ )H 1 (u, v) = 0 for a u, v. One can check from exampes (e.g. the C 4 degree 5 B-spine scheme) that 3 Υ 1 3 Υ. Consequenty, the term H 1 (u, v) = 0 for a u, v. Using the condition P 1 (u, v) = 0 to repace the term invoving F 0,3 in H 1 (u, v) with ower order expression shows that H 1 (u, v) = 1 3 [F 1,(v, u, v) F 1, (u, v, v) + F 0, (u, F 0, (v, v)) F 0, (v, F 0, (u, v))]. Consequenty, Equation (4.11) and Lemma impy that the connection induced by f has vanishing curvature. We have shown that 3 Φ 4,h+σ (u, v) = ( 3 Ξ Ξ 3 ) Q 1 (u) + ( 3 Ξ 3 3 Ξ 3 ) Q (u). We now know that the connection induced by f is torsion-free with vanishing curvature. It is a we-known fact in differentia geometry (see for instance [4]) that we can choose specia coordinates, centered at x 0, in which a connection coefficients Γ k ij = f (x,0) X i X vanish for x in a neighborhood of x j 0, i.e. We ca such coordinates fat coordinates. F 0, 0. By Remark 15, we may compute the proximity condition in fat coordinates. The initia vaue probem (.8) defining the exponentia map then reduces to the form Consequenty, the exponentia map is given by and the order k-condition reduces to F 0,k = 0. ẍ = 0, x(0) = x ẋ(0) = X. exp f (x, X) = x + X. Notice that in fat coordinates, F k, = 0 for a k, and therefore the condition P f = 0 reduces to the equivaent condition F 0,3 0. This in turn impies that F k,3 = 0 for a k. It foows that (in fat coordinates) Q 1 (u) = 0, and 3 Φ 4,h+σ (u, v) reduces to the expression 3 Φ 4,h+σ (u, v) = 1 4 ( 3 Ξ 3 3 Ξ 3 ) F 0,4 (u, u, u, u). One can check from exampes (e.g. the C 4 degree 5 B-spine scheme) that ( 3 Ξ 3 3 Ξ 3 ) 0. Hence, the C 4 -proximity condition impies F 0,4 0. But the condition that f satisfy the order 4 condition 9 aso reduces to F 0,4 0. Consequenty, (for schemes without dua time-symmetry) the vanishing curvature condition together with the order 4 condition, are equivaent to the 4-th order proximity condition 3 Φ 4,h+σ 0. 9 For the reference, the order 4 condition on a genera manifod can be expressed in a genera coordinate system as F0,4 u 4 F, (u ; u ) 4F 1, (u; u, F 0, u ) F 1, (F 0, u ; u ) F 0, (F 1, (u; u ), u) F 0, (F 0, (u ) ) 4F 0, (u, F 0, (u, F 0, (u ))) = 0, assuming that the order 3 condition P f (u) = F 0, (u, F 0, (u, u)) + 1 F 1,(u; u, u) 1 F 0,3(u, u, u) = 0 aready hods.