Modern Developments in the Theory and Applications of Moving Frames Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455 [email protected] http://www.math.umn.edu/∼olver Abstract. This article discusses recent advances in the general equivariant approachto the method of moving frames, concentrating on ﬁnite-dimensional Lie group actions. Afew of the many applications — to geometry, invariant theory, diﬀerential equations, andimage processing — are presented.† Supported in part by NSF Grant DMS 11–08894. September 4, 2014 1

1. Introduction. According to Akivis, [2], the method of rep`eres mobiles, which was translated intoEnglish as moving frames†, can be traced back to the moving trihedrons introduced bythe Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauß andLobachevsky. The apotheosis of the classical development can be found in the seminaladvances of E´lie Cartan, [21, 22], who forged earlier contributions by Cotton, Darboux,Frenet, Serret, and others into a powerful tool for analyzing the geometric properties ofsubmanifolds and their invariants under the action of transformation groups. An excel-lent English language treatment of the Cartan approach can be found in the book byGuggenheimer, [42]. The 1970’s saw the ﬁrst attempts, cf. [25, 39, 40, 53], to place Cartan’s construc-tions on a ﬁrm theoretical foundation. However, the method remained mostly constrainedwithin classical geometrical contexts, e.g. Euclidean, equi-aﬃne, or projective actions onsubmanifolds of Euclidean space and certain classical homogeneous spaces. In the late1990’s, I began to investigate how moving frames and all their remarkable consequencesmight be adapted to more general, non-geometrically-based group actions that arise inbroad range of compelling applications. The crucial conceptual leap was to decouple themoving frame theory from reliance on any form of frame bundle. Indeed, a careful studyof Cartan’s analysis of moving frames for curves in the projective plane, [21], in which hecalls a certain 3 × 3 unimodular matrix the “rep`ere mobile”, provided the crucial break-through, leading to the general, and universally applicable, deﬁnition of a moving frameas an equivariant map from the manifold or jet bundle back to the transformation group,thereby completely circumventing the many complications and diﬃculties inherent in the(higher order) frame bundle approach. Building on this basic idea, and armed with thepowerful tool of the variational bicomplex, [6, 127], Mark Fels and I, [31, 32], were ableto formulate a new, powerful, constructive equivariant moving frame theory that can besystematically applied to general transformation groups. All classical moving frames canbe reinterpreted in the equivariant framework, but the latter approach immediately appliesin far broader generality. Indeed, in later work with Pohjanpelto, [108, 109, 110, 111],the theory and algorithms were successfully extended to the vastly more complicated caseof inﬁnite-dimensional Lie pseudo-groups, [66, 67, 121]. Cartan’s normalization process for construction of the moving frame relies on thechoice of a cross-section to the group orbits, leading to a systematic construction of thefundamental invariants for the group action. Building on these two simple ideas, one mayalgorithmically construct equivariant moving frames and, as a result, complete systems of(diﬀerential) invariants for completely general group actions. The resulting moving frameconstruction induces a powerful invariantization process that associates each standard ob-ject (function, diﬀerential form, tensor, diﬀerential operator, variational problem, numeri-cal algorithm, etc.) with a canonically constructed invariant counterpart. Invariantization † According to my Petit Larousse, [68], the word “rep`ere” refers to a temporary mark madeduring building or interior design, and so a more faithful English translation might have been“movable landmarks”. 2

of the associated variational bicomplex produces the remarkable recurrence formulae, thatenable one to completely determine structure of the algebras of diﬀerential invariants, in-variant diﬀerential forms, invariant variational problems, invariant conservation laws, etc.,using only linear diﬀerential algebra, and, crucially, without having to know any explicitformulas for either the invariants or the moving frame itself ! It is worth emphasizing thatall of the constructions and required quantities can be constructed completely systemati-cally and algorithmically, and thus directly implemented in symbolic computer packages.Mansﬁeld’s recent text, [73], on what she calls the “symbolic invariant calculus”, pro-vides a basic introduction to the key ideas, albeit avoiding diﬀerential forms, and someof the important applications. An algebraically-based reformulation, adapted to symboliccomputation, has been proposed by Hubert and Kogan, [49, 50]. In general, the existence of a moving frame requires freeness of the underlying groupaction. Classically, non-free actions are made free by prolonging to jet space. Implemen-tation of the equivariant moving frame construction based on normalization through achoice of cross-section to the prolonged group orbits produces the fundamental diﬀerentialinvariants and, consequently, the solution to basic equivalence and symmetry problems forsubmanifolds via their diﬀerential invariant signature. Further, the equivariant movingframe calculus was also applied to Cartesian product actions, leading to complete classi-ﬁcations of joint invariants, joint diﬀerential invariants, and the associated joint invariantsignatures, [100]. Subsequently, a seamless amalgamation of jet and Cartesian productactions called multi-space was proposed† in [101] to serve as the basis for the geomet-ric analysis of numerical approximations, and, via the application of the moving framemethod, to the systematic construction of symmetry-preserving numerical approximationsand integration algorithms, [11, 19, 20, 58, 59, 60, 134]. With the basic moving frame machinery in hand, a plethora of new, unexpected,and signiﬁcant applications soon began appearing. In [9, 61, 98], the theory was appliedto produce new algorithms for solving the basic symmetry and equivalence problems ofpolynomials that form the foundation of classical invariant theory. The moving framemethod provides a direct route to the classiﬁcation of joint invariants and joint diﬀeren-tial invariants, [32, 100, 12], establishing a geometric counterpart of what Weyl, [135],in the algebraic framework, calls the ﬁrst main theorem for the transformation group.In [20, 11, 5, 8, 118, 92], the characterization of submanifolds via their diﬀerential invari-ant signatures was applied to the problem of object recognition and symmetry detection,[16, 17, 18, 30, 114]. The all-important recurrence formulae provide a complete charac-terization of the diﬀerential invariant algebra of group actions, and lead to new resultson minimal generating invariants, even in very classical geometries, [102, 47, 104, 51, 48].The general problem from the calculus of variations of directly constructing the invariantEuler-Lagrange equations from their invariant Lagrangians was solved in [63], and thenapplied, [103, 55, 130], to the analysis of the evolution of diﬀerential invariants under in- † Unfortunately, to date the fully rigorous multi-space construction is only known for curves,i.e., functions of a single variable. However, this theoretical diﬃculty does not prevent the appli-cation of the moving frame formalism to the practical design of numerical algorithms for partialdiﬀerential equations. 3

variant submanifold ﬂows, leading to integrable soliton equations and signature evolutionin computer vision. Applications of equivariant moving frames developed by other research groups includethe computation of symmetry groups and classiﬁcation of partial diﬀerential equations[72, 87]; symmetry and equivalence of polygons and point conﬁgurations, [13, 54]; ge-ometry and dynamics of curves and surfaces in homogeneous spaces, with applications tointegrable systems, Poisson geometry, and evolution of spinors, [75, 76, 77, 78, 79, 115];recognition of DNA supercoils, [117]; recovering structure of three-dimensional objectsfrom motion, [8]; classiﬁcation of projective curves in visual recognition, [43]; construc-tion of integral invariant signatures for object recognition in 2D and 3D images, [33]; thedesign and analysis of geometric integrators and symmetry-preserving numerical schemes,[26, 91, 116]; determination of invariants and covariants of Killing tensors and orthogo-nal webs, with applications to general relativity, separation of variables, and Hamiltoniansystems, [27, 29, 83, 84]; the Noether correspondence between symmetries and invariantconservation laws, [36, 37]; symmetry reduction of dynamical systems, [52, 120]; fur-ther developments in classical invariant theory, [62]; computation of Casimir invariants ofLie algebras and the classiﬁcation of subalgebras, with applications in quantum mechan-ics, [14, 15]. Applications to the study of the cohomology of the variational bicomplexand characteristic classes can be found in [124], generalizing earlier work on the pro-jectable case in [7]. Applications of the extension of the method to Lie pseudo-groups,[108, 109, 110, 111], including inﬁnite-dimensional symmetry groups of partial diﬀeren-tial equations, [23, 24, 88, 128]; to climate and turbulence modeling in [10]; to partialdiﬀerential equations arising in control theory in [129]; to the classiﬁcation of Laplaceinvariants and factorization of linear partial diﬀerential operators in [119]; to the con-struction of coverings and B¨acklund transformations, [89]; and to the method of groupfoliation, [133, 112], for ﬁnding invariant, partially invariant, and other explicit solutionsto partial diﬀerential equations, [123, 125]. In [131, 129, 85] the moving frame calculus isshown to provide a new and very promising alternative to the Cartan exterior diﬀerentialsystems approach, [34, 96], to solving a broad range of equivalence problems. Finally,recent generalizations to a theory of discrete equivariant moving frames have been appliedto integrable diﬀerential-diﬀerence systems, [74], and invariant evolutions of projectivepolygons, [80], that generalize the remarkable integrable pentagram maps, [56, 113].2. Equivariant Moving Frames. We begin by describing the general equivariant moving frame construction for ﬁnite-dimensional Lie group actions. Extensions to inﬁnite-dimensional Lie pseudo-groups aremore technically demanding, and we refer the interested reader to the survey paper [108]. The starting point is an r-dimensional Lie group G acting smoothly on an m-dimen-sional manifold M .Deﬁnition 2.1. A moving frame is a smooth, G-equivariant map ρ : M → G.There are two principal types of equivariance: g · ρ(z), left moving frame, (2.1)ρ(g · z) = ρ(z) · g−1, right moving frame. 4

z Oz g = ρ(z) k K Figure 1. Moving Construction Based on Cross–Section.In classical geometries, one can always reinterpret any classical frame-based moving frame,cf. [42], as a left-equivariant map. For example, the standard Euclidean moving frame fora space curve consists of three orthonormal vectors — the unit tangent, normal, andbinormal — as well as the point on the curve at which they are based, a fact routinelyignored but consistently emphasized by Cartan, [21], who calls it the “moving frame oforder 0”. If one interprets the orthonormal frame vectors as an orthogonal matrix andthe point on the curve as a translation vector, this eﬀectively deﬁnes a map from thecurve† to the Euclidean group E(3) = O(3) ⋉ R3. The resulting map is easily seen to beleft-equivariant, and hence satisfy the requirement of Deﬁnition 2.1. On the other hand, right-equivariant moving frames are often easier to compute, andwill be the primary focus here. Bear in mind that if ρ(z) is a right-equivariant movingframe, then ρ(z) = ρ(z)−1 is a left-equivariant counterpart. It is not diﬃcult to establish the basic requirements for the existence of an equivariantmoving frame. Theorem 2.2. A moving frame exists in a neighborhood of a point z ∈ M if andonly if G acts freely and regularly near z. Recall that G acts freely if the isotropy subgroup Gz = { g ∈ G | g · z = z } of eachpoint z ∈ M is trivial: Gz = {e}. This implies local freeness, meaning that the isotropysubgroups Gz are all discrete, or, equivalently, that the orbits all have the same dimension,r, as G itself. Regularity requires that, in addition, the orbits form a regular foliation; itis a global condition that plays no role in practical applications. † Or, more accurately, the second order jet of the curve, since they depend upon second orderderivatives. 5

The explicit construction of a moving frame is based on Cartan’s normalization pro-cedure. It is based on the choice of a (local) cross-section to the group orbits, meaning an(m − r)-dimensional submanifold K ⊂ M that intersects each orbit transversally and atmost once. Theorem 2.3. Let G act freely and regularly on M , and let K ⊂ M be a cross-section. Given z ∈ M , let g = ρ(z) be the unique group element that maps z to thecross-section: g · z = ρ(z) · z = k ∈ K. Then ρ : M → G is a right moving frame. The normalization construction of the moving frame is illustrated in Figure 1. Thecurves represent group orbits, with Oz denoting the orbit through the point z ∈ M . Thecross-section K is drawn as if it is a coordinate cross-section, while k = ρ(z) · z, the uniquepoint in the intersection Oz ∩ K, can be viewed as the canonical form or normal form, asprescribed by the cross-section, of the point z. Introducing local coordinates z = (z1, . . . , zm) on M , suppose that the cross-sectionK is deﬁned by the r equationsZ1(z) = c1, . . . Zr(z) = cr, (2.2)where Z1, . . . , Zr are scalar-valued functions, while c1, . . . , cr are suitably chosen constants.In the vast majority of applications, the Zν are merely a subset of the coordinate func-tions z1, . . . , zm, in which case they are said to deﬁne a coordinate cross-section. (Indeed,Figure 1 is drawn as if K is a coordinate cross-section.) The associated right moving frameg = ρ(z) is obtained by solving the normalization equationsZ1(g · z) = c1, . . . Zr(g · z) = cr, (2.3)for the group parameters g = (g1, . . . , gr) in terms of the coordinates z = (z1, . . . , zm).Transversality combined with the Implicit Function Theorem implies the existence of alocal solution g = ρ(z) to these algebraic equations, with equivariance assured by Theo-rem 2.3. In practical applications, the art of the method is to select a cross-section thatsimpliﬁes the calculations as much as possible. With the moving frame in hand, the next step is to determine the invariants, that is,(locally deﬁned) functions on M that are unchanged by the group action: I(g · z) = I(z)for all z ∈ dom I and all g ∈ G such that g · z ∈ dom I. Equivalently, a function is invariantif and only if it is constant on the orbits. The speciﬁcation of a moving frame by choice of a cross-section induces a canonicalprocedure to map functions to invariants. Deﬁnition 2.4. The invariantization of a function F : M → R is the unique invariantfunction I = ι(F ) that coincides with F on the cross-section: I | K = F | K. In particular, if I is any invariant, then clearly ι(I) = I. Thus, invariantization deﬁnesa projection from the space of (smooth) functions to the space of invariants that, moreover,preserves all algebraic operations. Invariantization (and its many consequences) providesthe preeminent advantage of the equivariant approach over competing theories.6

Computationally, a function F (z) is invariantized by ﬁrst transforming it accordingto the group, F (g · z) and then replacing the group parameters by their moving frameformulae g = ρ(z), so that ι[ F (z) ] = F (ρ(z) · z). (2.4)In particular, invariantization of the coordinate functions yields the fundamental invari-ants: I1(z) = ι(z1), . . . , Im(z) = ι(zm). With these in hand, the invariantization of ageneral function F (z) is simply given by ι F (z1, . . . , zm) = F (I1(z), . . . , Im(z)). (2.5)In particular, the functions deﬁning the cross-section (2.2) have constant invariantization,ι(Zν (z)) = cν , and are known as the phantom invariants, leaving precisely m − r func-tionally independent basic invariants, in accordance with Frobenius’ Theorem, [95]. Thefact that invariantization does not aﬀect the invariants implies the elegant and powerfulReplacement Rule J (z1, . . . , zm) = J (I1(z), . . . , Im(z)), (2.6)that can be used to immediately rewrite any invariant J(z1, . . . , zm) in terms of the basicinvariants. In symbolic analysis, (2.6) is known as a rewrite rule, [49, 50], and underscoresthe advantages of the moving frame approach over other invariant-theoretic constructions,including Hilbert and Gro¨bner bases, [28], and inﬁnitesimal methods, [95]. Of course, most interesting group actions are not free, and therefore do not admitmoving frames in the sense of Deﬁnition 2.1. There are two classical methods that (usually)convert a non-free action into a free action. The ﬁrst is the Cartesian product action ofG on several copies of M ; application of the moving frame normalization constructionand invariantization produces joint invariants, [100]. The second is to prolong the groupaction to jet space, which is the natural setting for the traditional moving frame theory, andleads to diﬀerential invariants, [32]. Combining the two methods of jet prolongation andCartesian product results in joint diﬀerential invariants, [100], also known in the computervision literature as semi-diﬀerential invariants, [86, 132]. In applications of symmetrymethods in numerical analysis, one requires an amalgamation of all these actions into acommon framework, called multi-space, [101]. In this paper we will concentrate on thejet space mode of prolongation, and refer the interested reader to [107] for an overview ofother developments.3. Moving Frames on Jet Space and Diﬀerential Invariants. Given an action of the Lie group G on the manifold M , let us concentrate on its inducedaction on (embedded) submanifolds S ⊂ M of a ﬁxed dimension 1 ≤ p < m = dim M .Traditional moving frames are obtained by prolonging the group action to the nth order jetbundle Jn = Jn(M, p), which is deﬁned as the set of equivalence classes of p-dimensionalsubmanifolds under the equivalence relation of nth order contact at a single point; see[93, 96] for details. Since G maps submanifolds to submanifolds while preserving thecontact equivalence relation, it induces an action on the jet space Jn, known as its nth orderprolongation and denoted by G(n). In local coordinates — see below — the formulas for the 7

prolonged group action are straightforwardly found by implicit diﬀerentiation, althoughthe resulting expressions can rapidly become extremely unwieldy. We assume, without signiﬁcant loss of generality, that G acts eﬀectively on opensubsets of M , meaning that the only group element that ﬁxes every point in any openU ⊂ M is the identity element. This implies, [99], that the prolonged action is locallyfree on a dense open subset Vn ⊂ Jn for n ≫ 0 suﬃciently large, whose points z(n) ∈ Vnare known as regular jets. In all known examples that arise in applications, the prolongedaction is, in fact, free on such a Vn. However, recently, Scot Adams, [1], constructedrather intricate examples of smooth Lie group actions that do not become eventually freeon any open subset of the jet space. Indeed, Adams proves that if the group has compactcenter, the prolonged actions always become eventually free on an open subset of jetspace, whereas any connected Lie group with non-compact center admits actions that donot become eventually free. A real-valued function on jet space, F : Jn → R is known as a diﬀerential function†.A diﬀerential invariant is a diﬀerential function that is unaﬀected by the prolonged grouptransformations, so I(g(n) · z(n)) = I(z(n)) for all z(n) ∈ Jn and all g ∈ G such that bothz(n) and g(n) ·z(n) lie in the domain of I. Clearly, any functional combination of diﬀerentialinvariants is a diﬀerential invariant (on their common domain of deﬁnition) and thus wespeak, somewhat loosely, of the algebra of diﬀerential invariants associated with the actionof the transformation group on submanifolds of a speciﬁed dimension. Since diﬀerentialinvariants are often only locally deﬁned‡, to be fully rigorous, we should introduce thecategory of sheaves of diﬀerential invariants, [66]. However, since we concentrate entirelyon local results, this extra level of abstraction is unnecessary, and so we will leave theirsheaf-theoretic reformulation as a simple translational exercise for the experts. As above, the normalization construction based on a choice of local cross-sectionKn ⊂ Vn to the prolonged group orbits can be used to produce an nth order equivariantmoving frame ρ : Jn → G in a neighborhood of any regular jet. The cross-section Kn isprescribed by setting a collection of r = dim G independent nth order diﬀerential functionsto suitably chosen constantsZ1(z(n)) = c1, . . . Zr(z(n)) = cr. (3.1)As above, the associated right moving frame g = ρ(z(n)) is obtained by solving the nor-malization equationsZ1(g(n) · z(n)) = c1, . . . Zr(g(n) · z(n)) = cr, (3.2) † Throughout, functions, maps, etc., may only be deﬁned on an open subset of their indicatedsource space: dom F ⊂ Jn. Also, we identify F with its pull-backs, F ◦ πnk, under the standardjet projections πnk: Jk → Jn for any k ≥ n. A similar remark applies to diﬀerential forms on jetspace. ‡ On the other hand, in practical examples, diﬀerential invariants turn out to be algebraicfunctions deﬁned on Zariski open subsets of jet space, and so reformulating the theory in a morealgebro-geometric framework would be a worthwhile endeavor; see, for instance, [50]. 8

for the group parameters g = (g1, . . . , gr) in terms of the jet coordinates z(n). Oncethe moving frame is established, the induced invariantization process will map generaldiﬀerential functions F (z(k)), of any order k, to diﬀerential invariants I = ι(F ), whichare obtained by substituting the moving frame formulas for the group parameters in theirtransformed version: I (z(k)) = F (g(k) · z(k))|g=ρ(z(n)). (3.3)As before, invariantization preserves diﬀerential invariants, ι(I) = I, and hence deﬁnesa canonical projection (depending on the moving frame) from the algebra of diﬀerentialfunctions to the algebra of diﬀerential invariants. For calculations, we introduce local coordinates z = (x, u) on M , considering theﬁrst p components x = (x1, . . . , xp) as independent variables, and the latter q = m − pcomponents u = (u1, . . . , uq) as dependent variables. Submanifolds that are transverse tothe vertical ﬁbers { x = constant } can thus be locally identiﬁed as the graphs of functionsu = f (x). The splitting into independent and dependent variables induces correspondinglocal coordinates z(n) = (x, u(n)) on Jn, whose components uJα, for α = 1, . . . , q, and J =(j1, . . . , jk) a symmetric multi-index of order 0 ≤ k ≤ n with entries 1 ≤ jν ≤ p, representthe partial derivatives, ∂kuα/∂xj1 · · · ∂xjk , of the dependent variables with respect to theindependent variables, [95, 96]. Equivalently, we can identify the jet (x, u(n)) with the nthorder Taylor polynomial of the function at the point x — or, when n = ∞, which will beimportant below, as its Taylor series. The fundamental diﬀerential invariants are obtained by invariantization of the jetcoordinate functions:Hi = ι(xi), IJα = ι(uJα), α = 1, . . . , q, #J ≥ 0, (3.4)and we abbreviate (H, I(k)) = ι(x, u(k)) for those obtained from the jet coordinates oforder ≤ k. Keep in mind that the invariant IJα has order ≤ max{ #J, n}, where n isthe order of the moving frame, while Hi has order ≤ n. The fundamental diﬀerentialinvariants (3.4) naturally split into two classes. The r = dim G combinations deﬁning thecross-section (3.1) will be constant, and are known as the phantom diﬀerential invariants.(In particular, if G acts transitively on M and the moving frame is of minimal order, thenall the Hi and Iα are constant.) For k ≥ n, the remaining basic diﬀerential invariantsprovide a complete system of functionally independent diﬀerential invariants of order ≤ k.According to (2.5), the invariantization of a diﬀerential function F (x, u(k)) can besimply implemented by replacing each jet coordinate by the corresponding normalizeddiﬀerential invariant (3.4): ι F (x, u(k)) = F (H, I(k)). (3.5)In particular, the Replacement Rule, cf. (2.6), allows one to straightforwardly rewrite anydiﬀerential invariant J (x, u(k)) in terms the basic invariants: J (x, u(k)) = J (H, I(k)), (3.6)which thereby establishes their completeness. 9

The speciﬁcation of independent and dependent variables on M splits† the diﬀerentialone-forms on the submanifold jet space J∞ into horizontal forms, spanned by dx1, . . . , dxp,and contact forms, spanned by the basic contact forms p α = 1, . . . , q, 0 ≤ #J. (3.7)θJα = duJα − uαJ,i dxi, i=1In general, a diﬀerential one-form θ on Jn is called a contact form if and only if it isannihilated by all jets, so θ | jnS = 0 for all p-dimensional submanifolds S ⊂ M . Thissplitting induces a bigrading of the space of diﬀerential forms on J∞ where the diﬀerentialdecomposes into horizontal and vertical components: d = dH + dV , with dH increasingthe horizontal degree and dV the vertical (contact) degree. Closure, d ◦ d = 0, impliesthat dH ◦ dH = 0 = dV ◦ dV , while dH ◦ dV = − dV ◦ dH . The resulting structureis known as the variational bicomplex , [6, 63, 127], and lies at the heart of the geomet-ric/topological approach to diﬀerential equations, variational problems, symmetries andconservation laws, characteristic classes, etc., bringing powerful cohomological tools suchas spectral sequences, [82], to bear on analytical and geometrical problems. The invariantization process induced by a moving frame can also be applied to dif-ferential forms on jet space. Thus, given a diﬀerential form ω on Jk, its invariantizationι(ω) is the unique invariant diﬀerential form that agrees with ω when pulled back to thecross-section. As with diﬀerential functions, the invariantized form is found by ﬁrst trans-forming the form by the prolonged group action, and then replacing the group parametersby their moving frame formulae:ι(ω) = (g(k))∗ ω|g=ρ(z(n)). (3.8)An invariantized contact form remains a contact form, while an invariantized horizontalform is, in general, a combination of horizontal and contact forms. The complete collectionof invariantized diﬀerential forms serves to deﬁne the invariant variational bicomplex ,studied in detail in [63, 124]. For the purposes of analyzing the diﬀerential invariants, we can ignore the contactforms. (Although they are important in further applications, including invariant variationalproblems, [63], submanifold ﬂows, [103], and characteristic classes, [124].) We let πHdenote the projection that maps a one-form onto its horizontal component. The horizontalcomponents of the invariantized basis horizontal formsωi = πH (̟i), where ̟i = ι(dxi), i = 1, . . . , p. (3.9)form, in the language of [96], a contact-invariant coframe. The corresponding dual invari-ant diﬀerential operators D1, . . . , Dp are deﬁned by pp (3.10)dH F = (DiF ) dxi = (DiF ) ωi, i=1 i=1† The splitting only works at inﬁnite order. 10

for any diﬀerential function F , where D1, . . . , Dp are the usual total derivative operators,[95, 96]. In practice, the invariant diﬀerential operator Di is obtained by substituting themoving frame formulas for the group parameters into the corresponding implicit diﬀeren-tiation operator used to produce the prolonged group actions. The invariant diﬀerentialoperators map diﬀerential invariants to diﬀerential invariants, and hence can be iterativelyapplied to generate the higher order diﬀerential invariants. Indeed, it is known that allthe higher order diﬀerential invariants can be expressed as functions of a ﬁnite number ofgenerating diﬀerential invariants and their successive invariant derivatives; see Section 5for details. Example 3.1. The paradigmatic example is the action of the orientation-preservingEuclidean group SE(2), consisting of translations and rotations, on plane curves C ⊂ M =R2. The group transformation g = (φ, a, b) ∈ SE(2) maps the point z = (x, u) to the pointw = (y, v) = g · z, given by y = x cos φ − u sin φ + a, v = x sin φ + u cos φ + b. (3.11)The local coordinate formulas for the prolonged group transformations on the curve jetspaces Jn = Jn(M, 1) are obtained by successively applying the implicit diﬀerentiationoperator† 1 − ux Dy = cos φ sin φ Dx (3.12)to v, producingvy = Dy v = sin φ + ux cos φ , vyy = Dy2v = uxx , . . . (3.13) cos φ − ux sin φ (cos φ − ux sin φ)3It is not hard to see that the prolonged action is locally free on the entire ﬁrst order jetspace V1 = J1. (As in most treatments, we gloss over the remaining discrete ambiguitycaused by a 180◦ rotation; see [100] for a complete development.) The standard movingframe is based on the cross-section K1 = { x = u = ux = 0 }. (3.14)Solving the corresponding normalization equations y = v = vy = 0 for the group parame-ters produces the right moving frame φ = − tan−1 ux , a = − x + uux , b = xux − u , (3.15) 1 + u2x 1 + u2xwhich deﬁnes a locally right-equivariant map from J1 to SE(2), the ambiguity in theinverse tangent a consequence of the local freeness of the prolonged action. The classicalleft-equivariant Frenet frame, [42], is obtained by inverting the Euclidean group element(3.15), with resulting group parameters φ = tan−1 ux , a = x, b = u. (3.16)† The implicit diﬀerentiation operator is dual to the horizontal derivative dH Y = (cos φ − ux sin φ) dx; see (3.18) below. 11

Observe that the translation component ( a, b ) = (x, u) = z can be identiﬁed with thepoint on the curve (Cartan’s moving frame of order 0), while the columns of the rotationmatrix having angle φ are precisely the orthonormal frame vectors based at z ∈ C, therebyidentifying the left-equivariant moving frame with the classical construction, [42]. Invariantization of the jet coordinate functions is accomplished by substituting themoving frame formulae (3.15) into the prolonged group transformations (3.13), producingthe fundamental diﬀerential invariants:H = ι(x) = 0, I0 = ι(u) = 0, I1 = ι(ux) = 0,I2 = ι(uxx) = uxx , I3 = ι(uxxx) = (1 + ux2 )uxxx − 3 uxu2xx , (3.17) + ux2 )3/2 (1 + u2x)3 (1and so on. The ﬁrst three, corresponding to functions deﬁning the the cross-section (3.14),are the phantom invariants. The lowest order basic diﬀerential invariant is the Euclideancurvature: I2 = κ. The higher order diﬀerential invariants I3, I4, . . . will be identiﬁedbelow.Similarly, to invariantize the horizontal form dx, we ﬁrst apply a Euclidean transfor-mation: dy = cos φ dx − sin φ du = (cos φ − ux sin φ) dx − (sin φ) θ, (3.18)where θ = du − ux dx is the order zero basis contact form. Substituting the moving frameformulae (3.15) produces the invariant one-form ̟ = ι(dx) = 1 + u2x dx + ux θ. (3.19) 1 + u2xIts horizontal component ω = πH (̟) = 1 + u2x dx (3.20)is the usual arc length element, and is itself invariant modulo contact forms. The dualinvariant diﬀerential operator is the arc length derivative D= 1 ux2 Dx, (3.21) 1+which can be obtained directly by substituting the moving frame formulae (3.15) intothe implicit diﬀerentiation operator (3.12). As we will see, the higher order diﬀerentialinvariants can all be obtained by successively diﬀerentiating the basic curvature invariantwith respect to arc length.Example 3.2. Let n = 0, 1. In classical invariant theory, the planar actions y = αx + β , v = (γ x + δ)−nu, (3.22) γx + δof the general linear group G = GL(2) play a key role in the equivalence and symmetryproperties of binary forms, particularly when u = q(x) is a polynomial of degree ≤ n,[9, 44, 98], whose graph is viewed as a plane curve. 12

Since dy = dH y = ∆ dx, where σ = γ x + δ, ∆ = αδ − βγ, σ2the prolonged action is found by successively applying the dual implicit diﬀerentiationoperator σ2 ∆ Dy = Dx (3.23)to v, producing vy = σ ux − n γ u , vyy = σ2 uxx − 2 (n − 1) γ σ ux + n(n − 1) γ2u , ∆ σn−1 ∆2 σn−2 vyyy = σ3uxxx − 3 (n − 2)γ σ2uxx + 3 (n − 1)(n − 2)γ 2 σ ux − n(n − 1)(n − 2)γ3u , ∆3 σn−3and so on. The action is locally free on the regular subdomain V2 = {uH = 0} ⊂ J2, where H = u uxx − n − 1 u2x nis the classical Hessian covariant of u, cf. [98]. We can choose the cross-section deﬁnedby the normalizations y = 0, v = 1, vy = 0, vyy = 1.Solving for the group parameters gives the right moving frame formulae† α = u(1−n)/n √ , β = −x u(1−n)/n √ , H H γ = 1 u(1−n)/n ux , δ = u1/n − 1 x u(1−n)/nux. (3.24) n nSubstituting the normalizations (3.24) into the higher order transformation rules gives usthe diﬀerential invariants, the ﬁrst two of which are vyyy −→ J = T , vyyyy −→ K = V , (3.25) H 3/2 H2where the diﬀerential polynomialsT = u2uxxx −3 n− 2 u uxuxx + 2 (n − 1)(n − 2) u3x, n n2V = u3uxxxx − 4 n−3 u2uxuxxx + 6 (n − 2)(n − 3) u u2x uxx − 3 (n − 1)(n − 2)(n − 3) u4x, n n2 n3can be identiﬁed with classical covariants of the binary form u = q(x) obtained throughthe transvectant process, cf. [44, 98]. † See [9] for a detailed discussion of how to resolve the square root ambiguities caused bylocal freeness. 13

As in the Euclidean case, the higher order diﬀerential invariants can be written interms of the basic “curvature invariant” J and its successive invariant derivatives withrespect to the invariant diﬀerential operator D = u H−1/2Dx, (3.26)which is itself obtained by substituting the moving frame formulae (3.24) into the implicitdiﬀerentiation operator (3.23).4. Recurrence. While invariantization clearly respects all algebraic operations, it does not commutewith diﬀerentiation. A recurrence relation expresses a diﬀerentiated invariant in terms ofthe basic diﬀerential invariants — or, more generally, a diﬀerentiated invariant diﬀerentialform in terms of the normalized invariant diﬀerential forms. The recurrence relations arethe master key that unlocks the entire structure of the algebra of diﬀerential invariants,including the speciﬁcation of generators and the classiﬁcation of syzygies. Remarkably,they can be explicitly determined without knowing the actual formulas for either thediﬀerential invariants, or the invariant diﬀerential operators, or even the moving frame!Indeed, they follow, requiring only linear (diﬀerential) algebra, directly from the well-known and relatively simple formulas for the prolonged inﬁnitesimal generators for thegroup action, combined with the speciﬁcation of the cross-section normalizations. Let v1, . . . , vr be a basis for the inﬁnitesimal generators of our eﬀectively acting r-dimensional transformation group G, which we identify with a basis of its Lie algebra g.We prolong each inﬁnitesimal generator to Jn, resulting in the vector ﬁeldsvσ(n) = p ξσi (x, u) ∂ q n ϕJα,σ (x, u(k)) ∂ , σ = 1, . . . , r. (4.1) i=1 ∂xi k=#J =0 ∂ uαJ + α=1The coeﬃcients ϕJα,σ = vσ(n)(uαJ ) are calculated using the prolongation formula, [95, 96],ﬁrst written in the following explicit non-recursive form in [93]: pp ϕJα,σ = DJ ϕσα − ξσi uiα + ξσi uαJ,i, (4.2) i=1 i=1in which DJ = Dj1 · · · Djk are iterated total derivative operators, and uiα = ∂uα/∂xi. Given a moving frame on jet space, the universal recurrence relation for diﬀerentialinvariants takes the following form. As above, we let D1, . . . , Dp denote the invariantdiﬀerential operators associated with the prescribed moving frame.Theorem 4.1. Let F (x, u(k)) be a diﬀerential function and ι(F ) its moving frameinvariantization. Then r Di ι(F ) = ι Di(F ) + Riσ ι vσ(k)(F ) , (4.3) σ=1where R = { Riσ | i = 1, . . . , p, σ = 1, . . . , r } (4.4)are known as the Maurer–Cartan diﬀerential invariants. 14

The Maurer–Cartan invariants Riσ can, in fact, be characterized as the coeﬃcients ofthe horizontal components of the pull-backs of the Maurer–Cartan forms via the movingframe map ρ : Jn → G, [32]. In the particular case of curves, if G ⊂ GL(N ) is a matrix Liegroup, then the Maurer–Cartan invariants appear as the entries of the classical Frenet–Serret matrix Dρ(x, u(n))·ρ(x, u(n))−1, [42, 47, 77]. Explicitly, suppose µ1, . . . , µr ∈ g∗ arethe basis for the right-invariant Maurer–Cartan forms that is dual to the given Lie algebrabasis v1, . . . , vr ∈ g. Then the horizontal components of their pull-backs, νσ = ρ∗µσ, canbe expressed as a linear combination of the contact-invariant coframe (3.9), whereby pγσ = πH (νσ) = πH (ρ∗µσ) = Riσ ωi, σ = 1, . . . , r. (4.5) i=1 In practical calculations, one, in fact, does not need to know where the Maurer–Cartan invariants come from, or even what a Maurer–Cartan form is, since the Maurer–Cartan invariants can be directly determined from the recurrence formulae for the phantomdiﬀerential invariants, as prescribed by the cross-section (3.1). Namely, since ι(Zν ) = cνis constant, for each 1 ≤ i ≤ p, the phantom recurrence relations r0 = ι Di(Zν ) + Riσ ι vσ(n)(Zν ) , ν = 1, . . . , r, (4.6) σ=1form a system of r linear equations that, owing to the transversality of the cross-section,can be uniquely solved for the r Maurer–Cartan invariants Ri1, . . . , Rir. Substituting theresulting expressions back into the non-phantom recurrence relations leads to a completesystem of identities satisﬁed by the basic diﬀerential invariants, that fully characterizesthe structure of the resulting diﬀerential invariant algebra, [32, 102, 111]. Example 4.2. The prolonged inﬁnitesimal generators of the planar Euclidean groupaction on curve jets, as described in Example 3.1, arev1(n) = ∂x, v2(n) = ∂u,v3(n) = − u ∂x + x ∂u + (1 + ux2 ) ∂ux + 3 uxuxx ∂uxx + (4 uxuxxx + 3 ux2x) ∂uxxx + · · · ,where v1(n), v2(n) generate translations while v3(n) generates (prolonged) rotations. Ac-cording to (4.3), the invariant arc length derivative D = ι(Dx) of a diﬀerential invariantI = ι(F ) is speciﬁed by the recurrence relationD ι(F ) = ι(DxF ) + R1 ι v1(n)(F ) + R2 ι v2(n)(F ) + R3 ι v3(n)(F ) , (4.7)where R1, R2, R3 are the three Maurer–Cartan invariants. To determine their formulas,we write out (4.7) for the three phantom invariants:0 = D ι(x) = ι(1) + R1 ι(v1(n)(x)) + R2 ι(v2(n)(x)) + R3 ι(v3(n)(x)) = 1 + R1,0 = D ι(u) = ι(ux) + R1 ι(v1(n)(u)) + R2 ι(v2(n)(u)) + R3 ι(v3(n)(u)) = R2,0 = D ι(ux) = ι(uxx) + R1 ι(v1(n)(ux)) + R2 ι(v2(n)(ux)) + R3 ι(v3(n)(ux)) = κ + R3. 15

Solving the resulting linear system of equations, we ﬁndR1 = −1, R2 = 0, R3 = − κ = − I2. (4.8)Using these, the general recurrence relation (4.7) becomesD ι(F ) = ι(DxF ) − ι v1(n)(F ) − κ ι v3(n)(F ) . (4.9)In particular, the arc length derivatives of the basic invariants Ik = ι(uk) = ι(Dxku) aregiven by 1 k−1 k+1 2 jDIk = Ik+1 − I2 Ij Ik−j+1, (4.10) j=2of which the ﬁrst few are κs = DI2 = I3, DI4 = I5 − 10 I22I3, (4.11)κss = DI3 = I4 − 3 I23, DI5 = I6 − 15 I22 I4 − 10 I2 I32.These can be iteratively solved to produce the explicit formulae κ = I2, I2 = κ, κs = I3, I3 = κs, κss = I4 − 3 I23, κsss = I5 − 19 I22 I3, I4 = κss + 3 κ3, (4.12)κssss = I6 − 34 I22 I4 − 48 I2 I32 + 57 I25, I5 = κsss + 19 κ2κs, I6 = κssss + 34 κ2κss + 48 κ κs2 + 45 κ5,etc., relating the fundamental normalized and diﬀerentiated curvature invariants. The invariant diﬀerential operators map diﬀerential invariants to diﬀerential invari-ants. However, when dealing with submanifolds of dimension p ≥ 2, i.e., functions ofseveral variables, they do not necessarily commute, and so the order of diﬀerentiation isimportant. However, each commutator can be re-expressed as a linear combination thereof, p[ Dj, Dk ] = Dj Dk − Dk Dj = Yjik Di , (4.13) i=1where the coeﬃcients Yjik = − Ykij are themselves diﬀerential invariants, known as thecommutator invariants. The explicit formulas for the commutator invariants in termsof the fundamental diﬀerential invariants can also be found by recurrence, as we nowdemonstrate. Indeed, the recurrence relations (4.3) can be straightforwardly extended to invari-antized diﬀerential forms. Namely, if Ω is any diﬀerential form on Jn, then r (4.14) d ι(Ω) = ι(dΩ) + νσ ∧ ι[vσ(n)(Ω)], σ=1 16

where vσ(n)(Ω) denotes the Lie derivative of Ω with respect to the prolonged inﬁnitesimalgenerator, while νσ = ρ∗µσ are the pulled-back Maurer–Cartan forms. For our purposes,we only need to look at the case when Ω = dxi is a basis horizontal form, whereby rrd̟i = d ι(dxi) = ι(d2xi) + νσ ∧ ι(dξσi ) = νσ ∧ ι(dξσi ). σ=1 σ=1Ignoring the contact components, and using (3.10), (4.5), we are led to r pp dH ωi = Rjσ ι(Dkξσi ) ωj ∧ ωk. σ=1 j=1 k=1On the other hand, applying dH to (3.10) and then recalling (4.13), we ﬁnd p0 = dH2 F = dH (DiF ) ∧ ωi + (DiF ) dH ωi i=1 pp p= Dj(DkF ) ωj ∧ ωk + (DiF ) dH ωi j=1 k=1 i=1 p pr pp= Yjik (DiF ) ωj ∧ ωk + Rjσ ι(Dkξσi ) (DiF ) ωj ∧ ωk. i=1 j<k i=1 σ=1 j=1 k=1Since F is arbitrary, we can equate the individual coeﬃcients of (DiF ) ωj ∧ ωk, for j < k,to zero, thereby producing explicit formulae for the commutator invariants: rp Yjik = Rkσ ι(Dj ξσi ) − Rjσ ι(Dkξσi ) . (4.15) σ=1 j=15. The Algebra of Diﬀerential Invariants. As we have seen, any ﬁnite-dimensional group action admits an inﬁnite number offunctionally independent diﬀerential invariants of progressively higher and higher order.On the other hand, inﬁnite-dimensional pseudo-groups may or may not admit nontriv-ial diﬀerential invariants, depending upon how “large” they are. For example, both thepseudo-group of all local diﬀeomorphisms, or that of all local symplectomorphisms, [81],act transitively on a dense open subset of the jet space, and hence admit no non-constantlocal invariants. (Global invariants, such as Gromov’s symplectic capacity, [41], are not,at least as far as I know, amenable to moving frame techniques.) The Fundamental Basis Theorem states that the entire algebra of diﬀerential invari-ants can be generated from a ﬁnite number of low order invariants by repeated invariantdiﬀerentiation. In diﬀerential invariant theory, it assumes the role played by the algebraicHilbert Basis Theorem for polynomial ideals, [28]. 17

Theorem 5.1. Let G be a ﬁnite-dimensional Lie group or, more generally, a Liepseudo-group that acts eventually freely† on jets of p-dimensional submanifolds S ⊂M . Then, locally, there exist a ﬁnite collection of generating diﬀerential invariants I ={I1, . . . , Iℓ}, along with exactly p invariant diﬀerential operators D1, . . . , Dp, such that ev-ery diﬀerential invariant can be locally expressed as a function of the generating invariantsand their invariant derivatives DJ Iν = Dj1 Dj2 · · · Djk Iν, for ν = 1, . . . , l, 1 ≤ jν ≤ p,k = #J ≥ 0.The Basis Theorem was ﬁrst formulated by Lie, [71; p. 760], for ﬁnite-dimensionalgroup actions. Modern proofs of Lie’s result can be found in [96, 112]. The theoremwas extended to inﬁnite-dimensional pseudo-groups by Tresse, [126]. A rigorous version,based on the machinery of Spencer cohomology, was established by Kumpera, [66]. Arecent generalization to pseudo-group actions on diﬀerential equations (subvarieties of jetspace) can be found in [65], while [90] introduces an approach based on Weil algebras. Theﬁrst constructive proof of the pseudo-group version, based on the moving frame machinery,appears in [111].Keep in mind that, the invariant diﬀerential operators do not necessarily commute,cf. (4.13). Furthermore, the diﬀerentiated invariants DJ Iν are not necessarily functionallyindependent, but may be subject to certain functional relations or diﬀerential syzygies ofthe form H( . . . DJ Iν . . . ) ≡ 0. (5.1)The Syzygy Theorem, ﬁrst stated (not quite correctly) in [32] for ﬁnite-dimensional actions,and then rigorously formulated and proved in [111], states that there are a ﬁnite numberof generating diﬀerential syzygies. Again, this result can be viewed as the diﬀerentialinvariant algebra counterpart of the Hilbert Syzygy Theorem for polynomial ideals, [28]. It is worth pointing out that, since the prolonged vector ﬁeld coeﬃcients (4.2) arepolynomials in the jet coordinates uKβ of order #K ≥ 1, their invariantizations are poly-nomial functions of the fundamental diﬀerential invariants IKβ for #K ≥ 1. As a result,the diﬀerential invariant algebra is, typically, rational, and are thus amenable to analysisby adaptations of techniques from computational algebra, [28]. The precise requirementsare either that the group acts transitively on M , or, if intransitive, that, in some coordi-nate system, its inﬁnitesimal generators v1, . . . , vr depend rationally on the coordinatesz on M . For such transitive or inﬁnitesimally rational group actions, if the cross-sectionfunctions Z1, . . . , Zr depend rationally on the jet coordinates, then the Maurer–Cartaninvariants are rational functions of the basic invariants (H, I(n+1)), where n is the orderof the moving frame. Moreover, all the resulting recurrence formulae depend rationally onthe basic diﬀerential invariants, justifying the claim. The detailed structure theory of suchnon-commutative diﬀerential invariant algebras has not to date been investigated in anydetails, and would be a very worthwhile endeavor.Let us discuss what is and isn’t known in this regards. A ﬁnite set of diﬀerentialinvariants I = {I1, . . . , Il} is called generating if, locally, every diﬀerential invariant can † See [111] for an explanation of the technical “eventually free” requirement on pseudo-groups.Extending the Basis Theorem to non-free pseudo-group actions is a signiﬁcant open problem. 18

be expressed as a function of them and their iterated invariant derivatives: DJ Iν for0 ≤ #J. The Basis Theorem 5.1 says that ﬁnite generating sets of diﬀerential invariantsexist, and their determination is important for a range of applications. Let us present afew general results in this vein, followed by some speciﬁc examples — all consequences ofthe all-important recurrence relations.Let (5.2) I(k) = {H1, . . . , Hp} ∪ { IJα | α = 1, . . . , q, 0 ≤ #J ≤ k }denote the fundamental diﬀerential invariants arising from invariantization of the jet co-ordinates. In particular, assuming we choose a cross-section Kn ⊂ Jn that projects toa cross-section π0n(Kn) ⊂ M , then the invariants I(0) = { H1, . . . , Hp, I1, . . . Iq } are theordinary invariants for the action on M . If G acts transitively on M , the latter invariantsare all constant (phantom), and hence their inclusion in the following generating systemsis superﬂuous. The ﬁrst result on generating systems can be found in [32]. Theorem 5.2. If the moving frame has order n, then the set of fundamental diﬀer-ential invariants I(n+1) of order n + 1 forms a generating set.Proof : Since the phantom invariants have order ≤ n, solving the phantom recurrencerelations (4.6) for the Maurer–Cartan invariants implies that the latter have order ≤ n + 1.Let us rewrite the recurrence relation (4.3) for the basic diﬀerential invariant IJα = ι(uαJ )in the form rIJα,i = DiIJα − Riσ ϕαJ,σ(H, I(k)). (5.3) σ=1Consequently, provided k = #J ≥ n + 1, the left hand side is a basic diﬀerential invariantof order k + 1, while the right hand side depends on diﬀerential invariants of order ≤ k andtheir invariant derivatives. A simple reverse induction on k completes the proof. Q.E.D. In practice, the generating set presented in Theorem 5.2 contains many redundacies,and reducing its cardinality is an important problem. One approach is to rely on the factthat almost all practical moving frames are of “minimal order”, [102, 47]. Deﬁnition 5.3. A cross-section Kn ⊂ Jn is said to have minimal order if, for all0 ≤ k ≤ n, its projection Kk = πkn(Kn) forms a cross-section to the prolonged group orbitsin Jk. Theorem 5.4. Suppose the diﬀerential functions Z1, . . . , Zr deﬁne, as in (3.1), aminimal order cross-section. LetZ = { ι(Di(Zν)) | i = 1, . . . , p, ν = 1, . . . , r } (5.4)be the collected invariantizations of their total derivatives. Then I(0) ∪ Z form a generatingset of diﬀerential invariants. Another interesting consequence of the recurrence relations, noticed by Hubert, [48],is that the Maurer–Cartan invariants (4.4) also form a generating set when the action istransitive on M . More generally: 19

Theorem 5.5. The diﬀerential invariants I(0) ∪ R form a generating set.Proof : By induction, the recurrence relations (5.3) imply that, for any k = #J > 0,we can rewrite the diﬀerential invariants of order k + 1 in terms of derivatives of those oforder k and the Maurer–Cartan invariants. Q.E.D. Let us now discuss the problem of ﬁnding a minimal generating set of diﬀerentialinvariants. The case of curves, p = 1, has been well understood for some time. A Lie groupis said to act ordinarily, [96], if it acts transitively on M , and the maximal dimension of theorbits of its successive prolongations strictly increase until the action becomes locally free;or, in other words, its prolongations do not “pseudo-stabilize”, [97]. Almost all transitiveLie group actions are ordinary. For an ordinary action on curves in a m-dimensionalmanifold, there are precisely q = m − 1 generating diﬀerential invariants. Moreover,there are no syzygies among their invariant derivatives. Non-ordinary actions require oneadditional generator, and a single generating syzygy. On the other hand, when dealing with submanifolds of dimension p ≥ 2, i.e., functionsof more than one variable, there are, as yet, no general results on the minimal number ofgenerating diﬀerential invariants. And indeed, even in well-studied examples, the conven-tional wisdom on what diﬀerential invariants are required in a minimal generating set isoften mistaken. Example 5.6. Consider the action of the Euclidean group E(3) = O(3) ⋉ R3 onsurfaces S ⊂ R3. In local coordinates, we can identify (transverse) surfaces with graphsof functions u = f (x, y). The corresponding local coordinates on the surface jet bundleJn = Jn(R3, 2) are x, y, u, ux, uy, uxx, uxy, uyy, . . ., and, in general, ujk = ∂j+ku/∂xj∂ykfor j + k ≤ n. The prolonged Euclidean action is locally free on the regular subset V2 ⊂ J2consisting of second order jets of surfaces at non-umbilic points. The classical movingframe construction, [42], relies on the coordinate cross-sectionK2 = { x = y = u = ux = uy = uxy = 0, uxx = uyy }. (5.5)The resulting left moving frame consists of the point on the curve deﬁning the translationcomponent a = z ∈ R3, while the columns of the rotation matrix R = [ t1, t2, n ] ∈ O(3)contain the unit tangent vectors t1, t2 forming the Darboux frame to the surface, [42],along with the unit normal n. The fundamental diﬀerential invariants are denoted as Ijk = ι(ujk). In particular,κ1 = I20 = ι(uxx), κ2 = I02 = ι(uyy),are the principal curvatures; the moving frame is valid provided κ1 = κ2, meaning that weare at a non-umbilic point. The mean and Gaussian curvature invariantsH = 1 (κ1 + κ2), K = κ1κ2, 2are often used as convenient alternatives, since they eliminate some (but not all) of theresidual discrete ambiguities in the locally equivariant moving frame. Higher order dif-ferential invariants are obtained by diﬀerentiation with respect to the Frenet coframeω1 = πH ι(dx1), ω2 = πH ι(dx2), that diagonalizes the ﬁrst and second fundamental 20

forms, [42]. We let D1, D2 denote the dual invariant diﬀerential operators, which are inthe directions of the diagonalizing Darboux frame t1, t2. A basis for the inﬁnitesimal generators for the action on R3 is provided by the 6 vectorﬁelds v1 = − y ∂x + x ∂y, v2 = − u ∂x + x ∂u, v3 = − u ∂y + y ∂u, (5.6) w1 = ∂x, w2 = ∂y, w3 = ∂u,the ﬁrst three generating the rotations and the second three the translations. The recur-rence relations (4.3) of order ≥ 1 have the explicit form 3 D1Ijk = Ij+1,k + ϕjσk(0, 0, I(j+k))R1σ, σ=1 j + k ≥ 1. 3 D2Ijk = Ij,k+1 + ϕσjk(0, 0, I(j+k))R2σ, σ=1Here R1σ, R2σ, are the Maurer–Cartan invariants associated with the rotational group gen-erator vσ, while ϕjσk(0, 0, I(j+k)) = ι ϕjσk(x, y, u(j+k)) are its invariantized prolongationcoeﬃcients, obtained through (4.2). (The translational generators and associated Maurer–Cartan invariants only appear in the order 0 recurrence relations, and so, for our purposes,can be ignored.) In particular, the phantom recurrence relations of order > 0 are0 = D1I10 = I20 + R12, 0 = D2I10 = R22, (5.7)0 = D1I01 = R13, 0 = D2I01 = I02 + R23,0 = D1I11 = I21 + (I20 − I02)R11, 0 = D2I11 = I12 + (I20 − I02)R21.Solving these produces the Maurer–Cartan invariants:R11 = Y2, R12 = − κ1, R13 = 0, R21 = − Y1, R22 = 0, R23 = − κ2, (5.8) (5.9)where I12 D1κ2 I21 D2κ1 I20 − I02 κ1 − κ2 I02 − I20 κ2 − κ1 Y1 = = , Y2 = = ,the latter expressions following from the third order recurrence relations I30 = D1I20 = D1κ1, I21 = D2I20 = D2κ1, (5.10) I12 = D1I02 = D1κ2, I03 = D2I02 = D2κ2.The general commutator formula (4.15) implies that the Maurer–Cartan invariants (5.9)are also the commutator invariants: D1, D2 = D1 D2 − D2 D1 = Y2 D1 − Y1 D2. (5.11)Further, equating the two fourth order recurrence relations for I22 = ι(uxxyy), namely, D2I21 + I30I12 − 2 I122 + κ1κ22 = I22 = D1I12 − I21I03 − 2 I221 + κ12κ2, κ1 − κ2 κ1 − κ2 21

leads us to the celebrated Codazzi syzygyD22κ1 − D12κ2 + D1κ1 D1κ2 + D2κ1 D2κ2 − 2 (D1κ2)2 − 2 (D2κ1)2 − κ1κ2(κ1 − κ2) = 0. κ1 − κ2 (5.12)Using (5.9), we can, in fact, rewrite the Codazzi syzygy in the more succinct form K = κ1κ2 = − (D1 + Y1)Y1 − (D2 + Y2)Y2. (5.13)As noted in [42], the right hand side of (5.13) depends only on the ﬁrst fundamental formof the surface. Thus, the Codazzi syzygy (5.13) immediately implies Gauss’ TheoremaEgregium, that the Gauss curvature is an intrinsic, isometric invariant. Another directconsequence of (5.13) is the celebrated Gauss–Bonnet Theorem, [63]. Since we are dealing with a second order moving frame, Theorem 5.2 implies that thediﬀerential invariant algebra for Euclidean surfaces is generated by the basic diﬀerentialinvariants of order ≤ 3. However, (5.10) express the third order invariants as invariantderivatives of the principal curvatures κ1, κ2, and hence they, or, equivalently, the Gaussand mean curvatures H, K, form a generating system for the diﬀerential invariant algebra.This is well known. However, surprisingly, [104], neither is a minimal generating set! Herewe present a reﬁnement of this result. Deﬁnition 5.7. A surface S ⊂ R3 is mean curvature degenerate if, for any non-umbilic point z0 ∈ S, there exist scalar functions f1(t), f2(t), such that D1H = f1(H), D2H = f2(H), (5.14)at all points z ∈ S in a suitable neighborhood of z0. Clearly any constant mean curvature surface is mean curvature degenerate, withf1(t) ≡ f2(t) ≡ 0. Surfaces with non-constant mean curvature that admit a one-parametergroup of Euclidean symmetries, i.e., non-cylindrical or non-spherical surfaces of rotation,non-planar surfaces of translation, or helicoid surfaces, obtained by, respectively, rotating,translating, or screwing a plane curve, are also mean curvature degenerate since, by thesignature characterization of symmetry groups, [32], they have exactly one non-constantfunctionally independent diﬀerential invariant, namely their mean curvature H and henceany other diﬀerential invariant, including the invariant derivatives of H — as well as theGauss curvature K — must be functionally dependent upon H. There also exist surfaceswithout continuous symmetries that are, nevertheless, mean curvature degenerate sinceit is entirely possible that (5.14) holds, but the Gauss curvature remains functionally in-dependent of H. However, I do not know whether there is a good intrinsic geometriccharacterization of such surfaces, which are well deserving of further investigation. Theorem 5.8. If a surface is mean curvature nondegenerate then the algebra of dif-ferential invariants is generated entirely by the mean curvature and its successive invariantderivatives. 22

Proof : In view of the Codazzi formula (5.13), it suﬃces to write the commutatorinvariants Y1, Y2 in terms of the mean curvature. To this end, we note that the commutatoridentity (5.11) can be applied to any diﬀerential invariant. In particular, D1D2H − D2D1H = Y2 D1H − Y1 D2H, (5.15)and, furthermore, for j = 1 or 2, D1D2DjH − D2D1Dj H = Y2 D1Dj H − Y1 D2DjH. (5.16)Provided the nondegeneracy condition(D1H)(D2DjH) = (D2H)(D1DjH), for j = 1 or 2, (5.17)holds, we can solve (5.15–16) to write the commutator invariants Y1, Y2 as explicit rationalfunctions of invariant derivatives of H. Plugging these expressions into the right hand sideof the Codazzi identity (5.13) produces an explicit formula for the Gauss curvature as arational function of the invariant derivatives, of order ≤ 4, of the mean curvature, validfor all surfaces satisfying the nondegeneracy condition (5.17). Thus it remains to show that (5.17) is equivalent to mean curvature nondegeneracyof the surface. First, if (5.14) holds, then DiDjH = Difj(H) = fj′(H)DiH = fj′(H)fi(H), i, j = 1, 2.This immediately implies that(D1H)(D2DjH) = (D2H)(D1DjH), j = 1, 2, (5.18)proving mean curvature degeneracy. Vice versa, noting that, when restricted to the sur-face, since the contact forms all vanish, dH reduces to the usual diﬀerential, and hencethe degeneracy condition (5.18) implies that, for each j = 1, 2, the diﬀerentials dH andd(DjH) are linearly dependent everywhere on S. The general characterization theorem forfunctional dependency, [95; Theorem 2.16], thus implies that, locally, DjH can be writtenas a function of H, thus establishing the condition (5.14). Q.E.D. Similar results hold for surfaces in several other classical three-dimensional Klein ge-ometries, [51, 106]. Theorem 5.9. The diﬀerential invariant algebra of a generic surface S ⊂ R3 underthe standard action of • the centro-equi-aﬃne group SL(3) is generated by a single second order invariant; • the equi-aﬃne group SA(3) = SL(3) ⋉ R3 is generated by a single third order diﬀer- ential invariant, known as the Pick invariant, [122]; • the conformal group SO(4, 1) is generated by a single third order invariant; • the projective group PSL(4) is generated by a single fourth order invariant. Lest the reader be tempted at this juncture to make a general conjecture concerningthe diﬀerential invariants of surfaces in three-dimensional space, the following elementaryexample shows that, even for surfaces in R3, the number of generating invariants can bearbitrarily large.23

Example 5.10. Consider the abelian group actionz = (x, y, u) −→ x + a, y + b, u + p(x, y) , (5.19)where a, b ∈ R and p(x, y) is an arbitrary polynomial of degree ≤ n. In this case, forsurfaces u = f (x, y), the individual derivatives ujk with j + k ≥ n + 1 form a completesystem of independent diﬀerential invariants. The invariant diﬀerential operators are theusual total derivatives: D1 = Dx, D2 = Dy, which happen to commute. The higher orderdiﬀerential invariants are generated by diﬀerentiating the n + 1 diﬀerential invariants ujkof order n + 1 = j + k. Moreover, these invariants clearly form a minimal generating set,of cardinality n + 2. Complete local classiﬁcations of Lie group actions on plane curves and their associateddiﬀerential invariant algebras are known, [96]. Building on his complete (local) classiﬁ-cations of both ﬁnite-dimensional Lie groups, and inﬁnite-dimensional Lie pseudo-groups,acting on one- and two-dimensional manifolds, [70], Lie, in volume 3 of his monumentaltreatise on transformation groups, [69], exhibits a large fraction of the three-dimensionalclassiﬁcation. He claims to have completed it, but says there is not enough space to presentthe full details. As far as I know, the remaining calculations have not been found in hisnotes or personal papers. Later, Amaldi, [3, 4], lists what he says is the complete clas-siﬁcation. More recently, unaware of Amaldi’s papers, Komrakov, [64], asserts that sucha classiﬁcation is not possible since one of the branches contains an intractable algebraicproblem. Amaldi and Komrakov’s competing claims remain to be reconciled, although Isuspect that Komrakov is right. Whether or not the Lie–Amaldi classiﬁcation is complete,it would, nevertheless, be a worthwhile project to systematically analyze the diﬀerential in-variant algebras of curves and, especially, surfaces under each of the transformation groupsappearing in the Amaldi–Lie lists. Even with the powerful recurrence formulae at our disposal, the general problem ofﬁnding and characterizing a minimal set of generating diﬀerential invariants when the di-mension of the submanifold is ≥ 2 remains open. Indeed, I do not know of a veriﬁablecriterion for minimality, except in the trivial case when there is a single generating in-variant, let alone an algorithm that will produce a minimal generating set. It is worthpointing out that the corresponding problem for polynomial ideals — ﬁnding a minimalHilbert basis — appears to be intractable. However, the special structure of the diﬀerentialinvariant algebra prescribed by the form of the recurrence relations gives some reasons foroptimism that such a procedure might be possible.6. Equivalence and Signature. A motivating application of the moving frame method is to solve problems of equiv-alence and symmetry of submanifolds under group actions. Let us brieﬂy recall the keyconstructions and results, and then present a couple of applications in image processing. Given a group action of G on M , two submanifolds S, S ⊂ M are said to be equivalentif S = g · S for some g ∈ G. A symmetry of a submanifold is a self-equivalence, that is agroup transformation g ∈ G that maps S to itself: S = g·S. The solution to the equivalence 24

and symmetry problems for submanifolds is based on the functional interrelationshipsamong the fundamental diﬀerential invariants restricted to the submanifold. Suppose we have constructed an nth order moving frame ρ(n): Jn → G deﬁned on anopen subset of jet space. A submanifold S is called regular if its n-jet jnS lies in thedomain of deﬁnition of the moving frame. For any k ≥ n, we use J (k) = I(k) | jkS,where I(k) = ( . . . Hi . . . IJα . . . ), #J ≤ k, to denote the kth order restricted diﬀerentialinvariants .Deﬁnition 6.1. The kth order signature S(k) = S(k)(S) of the regular submanifoldS ⊂ M is the set parametrized by the restricted diﬀerential invariants J (k): jkS → Rnk , p+k .where nk = dim Jk =p+q k The submanifold S is called fully regular if J (k) has constant rank 0 ≤ tk ≤ p = dim Sfor all k ≥ n. In this case, S(k) forms an immersed submanifold of dimension tk — typicallywith self-intersections. In the fully regular case, tn < tn+1 < tn+2 < · · · < tl = tl+1 = · · · = t ≤ p, (6.1)where t is the diﬀerential invariant rank and l the diﬀerential invariant order of S. Theorem 6.2. Two fully regular p-dimensional submanifolds S, S ⊂ M are locallyequivalent if and only if they have the same diﬀerential invariant order l and their signaturemanifolds of order l + 1 are identical: S(l+1)(S) = S(l+1)(S). Since symmetries are merely self-equivalences, the signature also determines the sym-metry group of the submanifold. In particular, the dimension of the signature equals thecodimension of the symmetry group. Theorem 6.3. If S ⊂ M is a fully regular p-dimensional submanifold of diﬀerentialinvariant rank t, then its (local) symmetry group GS is an (r − t)–dimensional subgroupof G that acts locally freely on S. A submanifold with maximal diﬀerential invariant rank t = p, and hence only adiscrete symmetry group, is called nonsingular . The number of symmetries of a nonsingularsubmanifold is determined by its index , which is deﬁned as the number of points in S mapto a single generic point of its signature: ind S = min # (J (l+1))−1{σ} σ ∈ S(l+1) . (6.2) Theorem 6.4. If S is a nonsingular closed submanifold, then its symmetry group isa discrete subgroup of cardinality ind S. At the other extreme, a rank 0 or maximally symmetric submanifold, [105], has allconstant diﬀerential invariants, and so its signature degenerates to a single point. Thesecan, in fact, be algebraically characterized. Theorem 6.5. A regular p-dimensional submanifold S has diﬀerential invariant rank0 if and only if its symmetry group is a p-dimensional subgroup H ⊂ G and hence S is anopen submanifold of an H–orbit: S ⊂ H · z0. 25

Remark : “Totally singular” submanifolds, all of whose jets lie outside the regularsubsets of Jk for all k ≫ 0, may have even larger, non-free symmetry groups, but theseare not covered by the preceding results. See [99] for details, including Lie algebraiccharacterizations. Example 6.6. Specializing to the action of the Euclidean group SE(2) on planecurves, the Euclidean signature for a curve C ⊂ M = R2 is the planar curve S(C) ={ (κ, κs) } parametrized by the curvature invariant κ and its ﬁrst derivative with respect toarc length. Two fully regular planar curves are equivalent under an oriented rigid motionif and only if they have the same signature curve. The maximally symmetric curves have constant Euclidean curvature, and so theirsignature curve degenerates to a single point. These are the circles and straight lines, and,in accordance with Theorem 6.5, each is the orbit of its one-parameter symmetry subgroupof SE(2). The number of Euclidean symmetries of a closed nonsingular curve is equal to itsindex — the number of times the Euclidean signature is retraced as we traverse the originalcurve. An example of a Euclidean signature curve is displayed in Figure 2. The ﬁrst ﬁgureshows the curve, and the second its Euclidean signature; the axes in the signature plotare κ and κs. Note in particular the approximate three-fold symmetry of the curve isreﬂected in the fact that its signature has winding number three. If the symmetries wereexact, the signature would be exactly retraced three times on top of itself. The ﬁnal ﬁguregives a discrete approximation to the signature which is based on the invariant numericalalgorithms introduced in [20]. In Figure 4 we display some signature curves computed from a 70×70, 8-bit gray-scaleimage of a cross section of a canine heart, obtained from an MRI scan. The boundary ofthe ventricle has been automatically segmented through use of the conformally Rieman-nian moving contour or snake ﬂow [57, 137]. Underneath these images, we display theventricle boundary curve along with two successive smoothed versions obtained applicationof the standard Euclidean-invariant curve shortening ﬂow, [35, 38]. As the evolving curvesapproach circularity the signature curves exhibit less variation in curvature and appear tobe winding more and more tightly around a single point, which is the signature of a circleof area equal to the area inside the evolving curve. Despite the rather extensive smoothinginvolved, except for an overall shrinking as the contour approaches circularity, the basicqualitative features of the diﬀerent signature curves, and particularly their winding behav-ior, appear to be remarkably robust. See [55] for a theoretical justiﬁcation, based on themaximum principle for the induced parabolic ﬂow of the signature curve. Thus, the signature curve method has the potential to be of signiﬁcant practical usein the general problem of object recognition and symmetry classiﬁcation. It oﬀer severaladvantages over more traditional approaches. First, it is purely local, and therefore im-mediately applicable to occluded objects. Second, it provides a mechanism for recognizingsymmetries and approximate symmetries of the object. The design of a suitably robust“signature metric” for practical comparison of signatures is the subject of ongoing research,[45, 118]. In [46], the Euclidean-invariant signature was further reﬁned and applied to de-sign a Matlab program that automatically assembles jigsaw puzzles. An example, [136],appears in Figure 6; assembly takes under an hour on a standard Macintosh laptop. 26

Example 6.7. Let us ﬁnally look at the equivalence and symmetry problems for bi-nary forms, [94, 98]. According to the general moving frame construction in Example 3.2,the signature curve S = S(q) of a function (polynomial) u = q(x) is parametrized bythe covariants J2 and K, given in (3.25). As an immediate consequence of the generalequivalence Theorem 6.2, we establish a non-classical and surprisingly simple solution tothe equivalence problem for complex-valued binary forms. Theorem 6.8. Two nondegenerate complex-valued binary forms q(x) and q(x) areequivalent if and only if their signature curves are identical: S(q) = S(q). Thus, remarkably, the equivalence and symmetry properties of binary forms are en-tirely encoded by the functional relations among two particular absolute rational covari-ants. Moreover, any equivalence map x = ϕ(x) must satsify the pair of rational equationsJ (x)2 = J (x)2, K(x) = K(x). (6.3)In particular, the theory guarantees that ϕ is necessarily a linear fractional transformation! Furthermore, the symmetries of a nonsingular form can be explicitly determined bysolving the rational equations (6.3) where J = J and K = K. As a consequence ofTheorems 6.4 and 6.5, we are led to a complete characterization of the symmetry groupsof binary forms. See [9] for a Maple implementation of this method for computing discretesymmetries and classiﬁcation of univariate polynomials. Theorem 6.9. A nondegenerate binary form q(x) is maximally symmetric if andonly if it satisﬁes the following equivalent conditions: • q is complex-equivalent to a monomial xk, with k = 0, n. • The covariant T 2 is a constant multiple of H3 ≡ 0. • The signature is just a single point. • q admits a one-parameter symmetry group. • The graph of q coincides with the orbit of a one-parameter subgroup of GL(2).On the other hand, the binary form is nonsingular if and only if it is not complex-equivalentto a monomial if and only if it has a ﬁnite symmetry group. In her thesis, Kogan, [61], extends these results to forms in several variables. Inparticular, a complete signature for ternary forms, [62], leads to a practical algorithm forcomputing discrete symmetries of, among other cases, elliptic curves.27

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1 0.5 -0.5 0.5 1 -0.5 The Original Curve 2 2 1 1 0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0.25 0.5 0.75 1 1.25 1.5 1.75 2 -1 -1 -2 -2Euclidean Signature Curve Discrete Euclidean Signature4422 0.5 1 1.5 2 2.5 0.5 1 1.5 2-2 -2-4 -4-6 Discrete Aﬃne SignatureAﬃne Signature CurveFigure 2. The Curve x = cos t + 1 cos2 t, y = sin t + 1 sin2 t. 5 10 36

1 0.5 -0.5 0.5 1 -0.5 -1 The Original Curve7.5 7.55 52.5 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4-2.5 -2.5-5 -5-7.5 -7.5Euclidean Signature Curve Discrete Euclidean Signature4422 0.5 1 1.5 2 2.5 0.5 1 1.5 2-2 -2-4 -4-6 Discrete Aﬃne SignatureAﬃne Signature CurveFigure 3. The Curve x = cos t + 1 cos2 t, y = 1 x + sin t + 1 sin2 t. 5 2 10 37

Original Canine Heart Blow Up of the Left Ventricle MRI ImageBoundary of Left Ventricle 60 50 40 30 20 10 20 30 40 50 60 Original Contour 0.06 0.06 0.04 0.04 0.02 0.02-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06Discrete Euclidean Signature Smoothly Connected Euclidean Signature Figure 4. Canine Left Ventricle Signature. 38

60 0.0650 0.0440 0.023020 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 10 20 30 40 50 60 -0.02 -0.04 -0.0660 0.0650 0.0440 0.023020 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 10 20 30 40 50 60 -0.02 -0.04 -0.0660 0.0650 0.0440 0.023020 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 10 20 30 40 50 60 -0.02 -0.04 -0.06Figure 5. Smoothed Canine Left Ventricle Signatures. 39

Figure 6. The Baﬄer Jigsaw Puzzle. 40

# Modern Developmentsin the Theory and Applicationsof ...

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**Description: ** Modern Developmentsin the Theory and Applicationsof MovingFrames Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455

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