Equivalence and Invariants: an Overview

Transcription

1 Equivalence and Invariants: an Overview Peter J. Olver University of Minnesota olver Montevideo, 2014

2 The Basic Equivalence Problem M smoothm-dimensional manifold. G transformation group acting on M finite-dimensional Lie group infinite-dimensional Lie pseudo-group

3 Equivalence: Determine when two p-dimensional submanifolds are congruent: N and N M N = g N for g G Symmetry: Find all symmetries, i.e., self-equivalences or self-congruences: N = g N

4 Classical Geometry F. Klein Euclidean group: G = SE(m) =SO(m) R m E(m) =O(m) R m z A z + b A SO(m) oro(m), b R m, z R m isometries: rotations, translations, (reflections) Equi-affine group: G =SA(m) =SL(m) R m A SL(m) volume-preserving Affine group: G =A(m) =GL(m) R m A GL(m) Projective group: G =PSL(m +1) acting on R m RP m = Applications in computer vision

5 Tennis, Anyone?

6 Classical Invariant Theory Binary form: Q(x) = n k=0 ( ) n a k k x k Equivalence of polynomials (binary forms): Q(x) =(γx + δ) n Q ( ) αx + β γx + δ g = ( α β γ δ ) GL(2) multiplier representation of GL(2) modular forms

7 Q(x) =(γx + δ) n Q ( ) αx + β γx + δ Transformation group: g :(x, u) ( αx + β γx + δ, ) u (γx + δ) n Equivalence of functions equivalence of graphs Γ Q = { (x, u) =(x, Q(x)) } C 2

8 Calculus of Variations L(x, u, p)dx = L( x, ū, p)d x Standard Equivalence: L = LD x x = L Divergence Equivalence: ( x x + p x u L = LD x x + D x B )

9 Allowed Changes of Variables Fiber-preserving transformations = Lie pseudo-groups x = ϕ(x) ū = ψ(x, u) p = χ(x, u, p) = αp+ β δ Point transformations x = ϕ(x, u) ū = ψ(x, u) p = χ(x, u, p) = αp+ β γp+ δ α = ϕ u Contact transformations β = ϕ x γ = ϕ u δ = ϕ x x = ϕ(x, u, p) ū = ψ(x, u, p) p = χ(x, u, p) dū pd x = λ(du pdx) λ 0

10 Ordinary Differential Equations d 2 u dx 2 = F ( x, u, du dx ) = d2 ū d x 2 = F ( x, ū, dū d x = Reduce an equation to a solved form, e.g., linearization, Painlevé,... ) dx dt Control Theory d2 x = F (t, x, u) = = F ( t, x, ū ) dt2 Equivalence map: x = ϕ(x) ū = ψ(x, u) = Feedback linearization, normal forms,...

11 Linear o.d.e.: D[u] = 0 Differential Operators D = n i =0 Eigenvalue problem: D[u] =λu a i (x)d i Evolution or Schrödinger equation: u t = D[u] D ψ Equivalence map: x = ϕ(x) ū = ψ(x)u D = 1 ψ D ψ = exactly and quasi-exactly solvable quantum operators,...

12 Equivalence & Invariants Equivalent submanifolds N N must have the same invariants: I = I. Constant invariants provide immediate information: e.g. κ =2 κ =2 Non-constant invariants are not useful in isolation, because an equivalence map can drastically alter the dependence on the submanifold parameters: e.g. κ = x 3 versus κ = sinh x

13 Equivalence & Invariants Equivalent submanifolds N N must have the same invariants: I = I. Constant invariants provide immediate information: e.g. κ =2 κ =2 Non-constant invariants are not useful in isolation, because an equivalence map can drastically alter the dependence on the submanifold parameters: e.g. κ = x 3 versus κ = sinh x

14 Equivalence & Invariants Equivalent submanifolds N N must have the same invariants: I = I. Constant invariants provide immediate information: e.g. κ =2 κ =2 Non-constant invariants are not useful in isolation, because an equivalence map can drastically alter the dependence on the submanifold parameters: e.g. κ = x 3 versus κ = sinh x

15 However, a functional dependency or syzygy among the invariants is intrinsic: e.g. κ s = κ 3 1 κ s = κ 3 1 Universal syzygies Gauss Codazzi Distinguishing syzygies. Theorem. (Cartan) Two submanifolds are (locally) equivalent if and only if they have identical syzygies among all their differential invariants.

16 However, a functional dependency or syzygy among the invariants is intrinsic: e.g. κ s = κ 3 1 κ s = κ 3 1 Universal syzygies Gauss Codazzi Distinguishing syzygies. Theorem. (Cartan) Two submanifolds are (locally) equivalent if and only if they have identical syzygies among all their differential invariants.

17 However, a functional dependency or syzygy among the invariants is intrinsic: e.g. κ s = κ 3 1 κ s = κ 3 1 Universal syzygies Gauss Codazzi Distinguishing syzygies. Theorem. (Cartan) Two submanifolds are (locally) equivalent if and only if they have identical syzygies among all their differential invariants.

18 Finiteness of Generators and Syzygies There are, in general, an infinite number of differential invariants and hence an infinite number of syzygies must be compared to establish equivalence. But the higher order syzygies are all consequences of a finite number of low order syzygies!

19 Example Plane Curves C R 2 G transitive,ordinary Lie group action (no pseudo-stabilization) κ ds unique(uptofunctionsthereof)differentialinvariantof lowest order curvature unique (up to multiple) contact-invariant one-form of lowest order arc length element Theorem. Every differential invariant of plane curves under ordinary Lie group actions is a function of the curvature invariant and its derivatives with respect to arc length: I = F (κ, κ s,κ ss,...,κ m )

20 Orbits If κ is constant, then all the higher order differential invariants are also constant: κ = c, 0=κ s = κ ss = Theorem. κ is constant if and only if the curve is a (segment of) an orbit of a one-parameter subgroup. Euclidean plane geometry: G =E(2) circles,lines Equi-affine plane geometry: G =SA(2) conicsections Projective plane geometry: G =PSL(2) W curves (Lie & Klein)

21 Suppose κ is not constant, and assume κ s 0. Then every syzygy is, locally, equivalent to one of the form If we know d m κ ds m = H m (κ) m =1, 2, 3,... κ s = H 1 (κ) then we can determine all higher order syzygies: κ ss = d ds H 1 (κ) =H 1 (κ) κ s = H 1 (κ) H 1 (κ) H 2 (κ) and similarly for κ sss,etc. ( )

22 Consequently, all the higher order syzygies are generated by the fundamental first order syzygy κ s = H 1 (κ) ( ) For plane curves under an ordinary transformation group, we need only know a single syzygy between κ and κ s in order to establish equivalence!

23 Reconstruction When H 1 0, the generating syzygy equation κ s = H 1 (κ) ( ) is an example of an automorphic differential equation, meaning that it admits G as a symmetry group, and, moreover, all solutions are obtained by applying group transformations to a single fixed solution: u = g u 0 = Rob Thompson s 2013 thesis.

24 Example. The Euclidean syzygy equation κ s = H 1 (κ) ( ) is the following third order ordinary differential equation: (1 + u 2 x )u xxx 3u ( ) x u2 xx u (1 + u 2 = H xx 1 x )3 (1 + u 2 x )3/2 It admits G =SE(2)asasymmetrygroup. If H 1 0, then given any one solution u = f 0 (x), every other solution is obtained by applying a rigid motion to its graph. On the other hand, if H 1 0, then the solutions are all the circles and straight lines, being the graphs of one-parameter subgroups. Question for the audience: SE(2) is a 3 parameter Lie group, but the initial data (x 0,u 0,u 0 x,u0 xx )for( ) dependsupon4arbitraryconstants. Reconcile these numbers.

25 The Signature Map In general, the generating syzygies are encoded by the signature map σ : N R l of the submanifold N, which is parametrized by a finite collection of fundamental differential invariants: The image σ(x) =(I 1 (x),...,i l (x)) Σ =Imσ R l is the signature subset (or submanifold) of N.

26 Equivalence & Signature Theorem. Two regular submanifolds are equivalent N = g N if and only if their signatures are identical Σ = Σ

27 Signature Curves Definition. The signature curve Σ R 2 of a curve C R 2 under an ordinary transformation group G is parametrized by the two lowest order differential invariants: {( Σ = κ, dκ )} R 2 ds Theorem. Two regular curves C and C are equivalent: C = g C if and only if their signature curves are identical: S = S

28 Signature Curves Definition. The signature curve Σ R 2 of a curve C R 2 under an ordinary transformation group G is parametrized by the two lowest order differential invariants: {( Σ = κ, dκ )} R 2 ds Theorem. Two regular curves C and C are equivalent: C = g C for g G if and only if their signature curves are identical: Σ = Σ

29 Other Signatures Euclidean space curves: C R 3 κ curvature, τ torsion Σ = { ( κ, κ s,τ)} R 3 Euclidean surfaces: S R 3 (generic) H meancurvature, K Gausscurvature Σ = {( )} H,K,H,1,H,2,K,1,K,2 R 6 Σ = {( )} H,H,1,H,2,H,1,1 R 4 Equi affine surfaces: S R 3 (generic) P Pickinvariant Σ = {( )} P,P,1,P,2,P,1,1 R 3

30 Symmetry and Signature Theorem. group The dimension of the (local) symmetry G N = { g g N = N } of a nonsingular submanifold N M equals the codimension of its signature: dim G N = dim N dim Σ Corollary. For a nonsingular submanifold N M, 0 dim G N dim N = Totally singular submanifolds can have larger symmetry groups!

31 Maximally Symmetric Submanifolds Theorem. The following are equivalent: The submanifold N has a p-dimensional symmetry group The signature Σ degenerates to a point: dim Σ =0 The submanifold has all constant differential invariants N = H {z 0 } is the orbit of a(nonsingular)p-dimensional subgroup H G

32 Discrete Symmetries Definition. The index of a submanifold N equals the number of points in N which map to a generic point of its signature: ι N = min { # σ 1 {w} } w Σ = Self intersections Theorem. The number of local symmetries of a submanifold at a generic point z N equals its index ι z. = Approximate symmetries

33 The Index σ N Σ

34 The Curve x =cost cos2 t, y =sint sin2 t The Original Curve Euclidean Signature Equi-affine Signature

35 The Curve x =cost cos2 t, y = 1 2 x +sint sin2 t The Original Curve Euclidean Signature Equi-affine Signature

36 = Steve Haker

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39 The Baffler Jigsaw Puzzle

40 The Baffler Solved = Dan Hoff

41 Distinguishing Melanomas from Moles Melanoma Mole = A. Grim, A. Rodriguez, C. Shakiban, J. Stangl

42 Classical Invariant Theory M = R 2 \{u =0} G =GL(2)= {( α β γ δ ) =αδ βγ 0 } (x, u) ( αx + β γx + δ, ) u (γx + δ) n n 0, 1

43 Differential invariants: Hessian: κ = T 2 H 3 κ s U H 2 = absolute rational covariants H = 1 2 (u, u)(2) = n(n 1)uu xx (n 1) 2 u 2 x 0 Higher transvectants (Jacobian determinants): T =(u, H) (1) =(2n 4)u x H nu H x U =(u, T ) (1) =(3n 6)Q x T nq T x Theorem. Two nonsingular binary forms are equivalent if and only if their signature curves, parametrized by (κ, κ s ), are identical.

44 Symmetries of Binary Forms Theorem. The symmetry group of a nonzero binary form Q(x) 0ofdegreen is: A two-parameter group if and only if H 0ifandonlyif Q is equivalent to a constant. = totally singular A one-parameter group if and only if H 0andT 2 = ch 3 if and only if Q is complex-equivalent to a monomial x k,withk 0,n. = maximally symmetric In all other cases, a finite group whose cardinality equals the index of the signature curve, and is bounded by { 6n 12 U = ch 2 ι Q 4n 8 otherwise

45 Cartan s Main Idea Recast the equivalence problem for submanifolds under a (pseudo-)group action, in the geometric language of differential forms. Then reduce the equivalence problem to the most fundamental equivalence problem: Equivalence of coframes.

46 Coframes Let M be an m-dimensional manifold, e.g., M R m. Definition. A coframe on M is a linearly independent system of one-forms θ = {θ 1,...,θ m } forming a basis for its cotangent space T M z at each point z M. In other words θ i = m j =1 h i j (x) dxj, det( h i j (x)) 0

47 Equivalence of Coframes Definition. Two coframes θ on M and θ on M are equivalent if there is a diffeomorphism Φ : M M such that Φ θ i = θ i i =1,...,m

48 Since the exterior derivative d commutes with pull-back, Φ (dθ i )=dθ i i =1,...,m Structure equations dθ i = j<k I i jk θj θ k = The torsion coefficients are invariant: I i jk ( x) =Ii jk (x)

49 Covariant derivatives df = F θ 1 θ1 + + F θ m θm If I j is invariant, soareallitsderived invariants: I j,k = I j θ k I j,k,l = I j,k θ l... We now have a potentially infinite collection of invariants!

50 Rank and Order of a Coframe r n =#functionallyindependentinvariantsoforder n: r 0 =rank{i j } r 1 =rank{i j,i j,k }... r 0 <r 1 < <r s = r s+1 = r s+2 = Order = s Rank = r = r s

51 The Order 0 Case s =0 r = r 0 = r 1 = Syzygies: I j,k = F jk (I 1,...,I r ) Signature: parametrized by I j,i j,k.

52 Equivalence of Coframes Cartan s Theorem: Two order 0 coframes are equivalent if and only if Their ranks are the same Their signature manifolds are identical The invariant equations I j ( x) =I j (x) have a common real solution. Any solution to the invariant equations determines an equivalence between the two coframes.

53 Symmetry Groups of Coframes Theorem. Let θ be an invariant coframe of rank r on an m-dimensional manifold M. Thenθ admits an (m r)-dimensional (local) symmetry group.

54 Cartan s Graphical Proof Technique The graph of the equivalence map ψ : M M can be viewed as a transverse m-dimensional integral submanifold Γ ψ M M for the involutive differential system generated by the one-forms and functions θ i θ i I j I j Existence of suitable integrable submanifolds determining equivalence maps is guaranteed by the Frobenius Theorem, which is, at its heart, an existence theorem for ordinary differential equations, and hence valid in the smooth category.

55 Extended Coframes Definition. An extended coframe {θ, J } on M consists of acoframeθ = {θ 1,...,θ m } and acollectionoffunctionsj =(J 1,...,J l ). Two extended coframes are equivalent if there is a diffeomorphism Φ such that Φ θ k = θ k Φ J i = J i The solution to the equivalence of extended coframes is a straightforward extension of that of coframes. One merely adds the extra invariants J i to the collection of torsion invariants I i jk to form the basic invariants, and then applies covariant differentiation to all of them to produce the higher order invariants.

56 Determining the Invariant (Extended) Coframe There are now two methods for explicitly determining the invariant (extended) coframe associated with a given equivalence problem. The Cartan Equivalence Method Equivariant Moving Frames Either will produce the fundamental differential invariants required to construct a signature and thereby effectively solve the equivalence problem. Infinitesimal methods (solve PDEs)

57 The Cartan Equivalence Method (1) Reformulate the problem as an equivalence problem for G-valued coframes, for some structure group G (2) Calculate the structure equations by applying d (3) Use absorption of torsion to determine the essential torsion (4) Normalize the group-dependent essential torsion coefficients to reduce the structure group (5) Repeat the process until the essential torsion coefficients are all invariant (6) Test for involutivity (7) If not involutive, prolong (à la EDS) and repeat until involutive The result is an invariant coframe that completely encodes the equivalence problem, perhaps on some higher dimensional space. The structure invariants for the coframe are used to parametrize thesignature.

58 Equivariant Moving Frames (1) Prolong (à la jet bundle) the (pseudo-)group action to the jet bundle of order n where the action becomes (locally) free (2) Choose a cross-section to the group orbits and solve the normalization equations to determine an equivariant moving frame map ρ :J n G (3) Use invariantization to determine the normalized differential invariants of order n +1 and invariant differential forms; invariant differential operators;... (4) Apply the recurrence formulae to determine higher order differential invariants, and the structure of the differential invariant algebra Step (4) canbedonecompletelysymbolically,usingonlylinearalgebra, independent of the explicit formulae in step (3)

59 The Recurrence Formulae The moving frame recurrence formulae enable one to determine the generating differential invariants and hence the invariants I 1,...,I l required for constructing a signature. The extended coframe used to prove equivalence consists of the pulled-back Maurer Cartan forms ν i = ρ (µ I )alongwiththe generating differential invariants I j and their differentials di j. = It is not (yet) known how to construct the recurrence formulae through the Cartan equivalence method! = See Francis Valiquette s recent paper for an alternative method for solving Cartan equivalence problems using the moving frame approach for Lie pseudo-groups.

60 The Basis Theorem Theorem. Given a Lie group (or Lie pseudo-group )actingon p-dimensional submanifolds, the corresponding differential invariant algebra I G is locally generated by a finite number of differential invariants I 1,...,I k and p invariant differential operators D 1,...,D p meaning that every differential invariant can be locally expressed as a function of the generating invariants and their invariant derivatives: D J I i = D j1 D j2 D jn I i. = Lie groups: Lie, Ovsiannikov, Fels O = Lie pseudo-groups: Tresse, Kumpera, Kruglikov Lychagin, Muñoz Muriel Rodríguez, Pohjanpelto O

61 Key Issues Minimal basis of generating invariants: I 1,...,I l Commutation formulae for the invariant differential operators: [ D j, D k ]= p i=1 Y i jk D i Syzygies (functional relations) among = Non-commutative differential algebra Φ(... D J I κ... ) 0 the differentiated invariants:

62 Recurrence Formulae D i ι(f )=ι(d i F ) + r κ=1 ι invariantization map F (x, u (n) ) differential function I = ι(f ) differential invariant Ri κ ι(v(n) κ (F )) D i total derivative with respect to x i D i = ι(d i ) invariant differential operator v (n) κ infinitesimal generators of prolonged action of G on jets Ri κ Maurer Cartan invariants (coefficients of pulled-back Maurer Cartan forms)

63 Recurrence Formulae D i ι(f )=ι(d i F ) + r κ=1 Ri κ ι(v(n) κ (F )) If ι(f )= c is a phantom differential invariant coming from the moving frame cross-section, then the left hand side of the recurrence formula is zero. The collection of all such phantom recurrence formulae form a linear algebraic system of equations that can be uniquely solved for the Maurer Cartan invariants R κ i. Once the Maurer Cartan invariants Ri κ are replaced by their explicit formulae, the induced recurrence relations completely determine the structure of the differential invariant algebra I G!

64 Euclidean Surfaces Euclidean group SE(3) = SO(3) R 3 acts on surfaces S R 3. For simplicity, we assume the surface is (locally) the graph of a function z = u(x, y) Infinitesimal generators: v 1 = y x + x y, v 2 = u x + x u, v 3 = u y + y u, w 1 = x, w 2 = y, w 3 = u. The translations w 1, w 2, w 3 will be ignored, as they play no role in the higher order recurrence formulae.

65 Cross-section (Darboux frame): x = y = u = u x = u y = u xy =0. Phantom differential invariants: ι(x) =ι(y) =ι(u) =ι(u x )=ι(u y )=ι(u xy )=0 Principal curvatures κ 1 = ι(u xx ), κ 2 = ι(u yy ) Mean curvature and Gauss curvature: H = 1 2 (κ 1 + κ 2 ), K = κ 1 κ 2 Higher order differential invariants invariantized jet coordinates: I jk = ι(u jk ) where u jk = j+k u x j y k Nondegeneracy condition: non-umbilic point κ 1 κ 2.

66 Algebra of Euclidean Differential Invariants Principal curvatures: κ 1 = ι(u xx ), κ 2 = ι(u yy ) Mean curvature and Gauss curvature: H = 1 2 (κ 1 + κ 2 ), K = κ 1 κ 2 Invariant differentiation operators: D 1 = ι(d x ), D 2 = ι(d y ) = Differentiation with respect to the diagonalizing Darboux frame. The recurrence formulae enable one to express the higher order differential invariants in terms of the principal curvatures, or, equivalently, the mean and Gauss curvatures, and their invariant derivatives: I jk = ι(u jk )= Φ jk (κ 1,κ 2, D 1 κ 1, D 2 κ 1, D 1 κ 2, D 2 κ 2, D 2 1 κ 1,... ) =Φ jk (H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... )

67 Algebra of Euclidean Differential Invariants Principal curvatures: κ 1 = ι(u xx ), κ 2 = ι(u yy ) Mean curvature and Gauss curvature: H = 1 2 (κ 1 + κ 2 ), K = κ 1 κ 2 Invariant differentiation operators: D 1 = ι(d x ), D 2 = ι(d y ) = Differentiation with respect to the diagonalizing Darboux frame. The recurrence formulae enable one to express the higher order differential invariants in terms of the principal curvatures, or, equivalently, the mean and Gauss curvatures, and their invariant derivatives: I jk = ι(u jk )= Φ jk (κ 1,κ 2, D 1 κ 1, D 2 κ 1, D 1 κ 2, D 2 κ 2, D 2 1 κ 1,... ) =Φ jk (H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... )

68 Recurrence Formulae ι(d i u jk )=D i ι(u jk ) 3 κ =1 I jk = ι(u jk ) normalized differential invariants R κ i R κ i Maurer Cartan invariants ϕ jk κ (0, 0,I(j+k) )=ι[ ϕ jk κ (x, y, u(j+k) )] ι[ ϕjk κ (x, y, u(j+k) )], j + k 1 invariantized prolonged infinitesimal generator coefficients. I j+1,k = D 1 I jk I j,k+1 = D 1 I jk 3 κ =1 3 κ =1 ϕ jk κ (0, 0,I(j+k) ) R κ 1 ϕ jk κ (0, 0,I(j+k) ) R κ 2

69 Recurrence Formulae ι(d i u jk )=D i ι(u jk ) 3 κ =1 I jk = ι(u jk ) normalized differential invariants R κ i R κ i Maurer Cartan invariants ϕ jk κ (0, 0,I(j+k) )=ι[ ϕ jk κ (x, y, u(j+k) )] ι[ ϕjk κ (x, y, u(j+k) )], j + k 1 invariantized prolonged infinitesimal generator coefficients. I j+1,k = D 1 I jk I j,k+1 = D 1 I jk 3 κ =1 3 κ =1 ϕ jk κ (0, 0,I(j+k) ) R κ 1 ϕ jk κ (0, 0,I(j+k) ) R κ 2

70 Prolonged infinitesimal generators: pr v 1 = y x + x y u y ux + u x uy 2u xy uxx +(u xx u yy ) uxy 2u xy uyy +, pr v 2 = u x + x u +(1+u 2 x ) u x + u x u y uy +3u x u xx uxx +(u y u xx +2u x u xy ) uxy +(2u y u xy + u x u yy ) uyy +, pr v 3 = u y + y u + u x u y ux +(1+u 2 y ) u y +(u y u xx +2u x u xy ) uxx +(2u y u xy + u x u yy ) uxy +3u y u yy uyy +. Phantom differential invariants: I jk = ι(u jk ) I 00 = I 10 = I 01 = I 11 =0 Principal curvatures: I 20 = κ 1 I 02 = κ 2

71 Prolonged infinitesimal generators: pr v 1 = y x + x y u y ux + u x uy 2u xy uxx +(u xx u yy ) uxy 2u xy uyy +, pr v 2 = u x + x u +(1+u 2 x ) u x + u x u y uy +3u x u xx uxx +(u y u xx +2u x u xy ) uxy +(2u y u xy + u x u yy ) uyy +, pr v 3 = u y + y u + u x u y ux +(1+u 2 y ) u y +(u y u xx +2u x u xy ) uxx +(2u y u xy + u x u yy ) uxy +3u y u yy uyy +. Phantom differential invariants: I jk = ι(u jk ) I 00 = I 10 = I 01 = I 11 =0 Principal curvatures: I 20 = κ 1 I 02 = κ 2

72 Phantom recurrence formulae: κ 1 = I 20 = D 1 I 10 R 2 1 = R2 1, 0=I 11 = D 1 I 01 R 3 1 = R3 1, I 21 = D 1 I 11 (κ 1 κ 2 )R 1 1 = (κ 1 κ 2 )R1 1, 0=I 11 = D 2 I 10 R 2 2 = R2 2, κ 2 = I 02 = D 2 I 01 R 3 2 = R3 2, I 12 = D 2 I 11 (κ 1 κ 2 )R 1 2 = (κ 1 κ 2 )R1 2. Maurer Cartan invariants: R1 1 = Y 1, R2 1 = κ 1, R3 1 =0, Commutator invariants: R 2 1 = Y 2, R2 2 =0, R2 3 = κ 2. Y 1 = I 21 κ 1 κ 2 = D 1 κ 2 κ 1 κ 2 Y 2 = I 12 κ 1 κ 2 = D 2 κ 1 κ 2 κ 1 [ D 1, D 2 ]=D 1 D 2 D 2 D 1 = Y 2 D 1 Y 1 D 2,

73 Phantom recurrence formulae: κ 1 = I 20 = D 1 I 10 R 2 1 = R2 1, 0=I 11 = D 1 I 01 R 3 1 = R3 1, I 21 = D 1 I 11 (κ 1 κ 2 )R 1 1 = (κ 1 κ 2 )R1 1, 0=I 11 = D 2 I 10 R 2 2 = R2 2, κ 2 = I 02 = D 2 I 01 R 3 2 = R3 2, I 12 = D 2 I 11 (κ 1 κ 2 )R 1 2 = (κ 1 κ 2 )R1 2. Maurer Cartan invariants: R1 1 = Y 1, R2 1 = κ 1, R3 1 =0, Commutator invariants: R 2 1 = Y 2, R2 2 =0, R2 3 = κ 2. Y 1 = I 21 κ 1 κ 2 = D 1 κ 2 κ 1 κ 2 Y 2 = I 12 κ 1 κ 2 = D 2 κ 1 κ 2 κ 1 [ D 1, D 2 ]=D 1 D 2 D 2 D 1 = Y 2 D 1 Y 1 D 2,

74 Phantom recurrence formulae: κ 1 = I 20 = D 1 I 10 R 2 1 = R2 1, 0=I 11 = D 1 I 01 R 3 1 = R3 1, I 21 = D 1 I 11 (κ 1 κ 2 )R 1 1 = (κ 1 κ 2 )R1 1, 0=I 11 = D 2 I 10 R 2 2 = R2 2, κ 2 = I 02 = D 2 I 01 R 3 2 = R3 2, I 12 = D 2 I 11 (κ 1 κ 2 )R 1 2 = (κ 1 κ 2 )R1 2. Maurer Cartan invariants: R1 1 = Y 1, R2 1 = κ 1, R3 1 =0, Commutator invariants: R 2 1 = Y 2, R2 2 =0, R2 3 = κ 2. Y 1 = I 21 κ 1 κ 2 = D 1 κ 2 κ 1 κ 2 Y 2 = I 12 κ 1 κ 2 = D 2 κ 1 κ 2 κ 1 [ D 1, D 2 ]=D 1 D 2 D 2 D 1 = Y 2 D 1 Y 1 D 2,

75 Third order recurrence relations: I 30 = D 1 κ 1 = κ 1,1, I 21 = D 2 κ 1 = κ 1,2, I 12 = D 1 κ 2 = κ 2,1, I 03 = D 2 κ 2 = κ 2,2, Fourth order recurrence relations: I 40 = κ 1,11 3κ2 1,2 κ 1 κ 2 +3κ 3 1, I 31 = κ 1,12 3κ 1,2 κ 2,1 κ 1 κ 2 = κ 1,21 + κ 1,1 κ 1,2 2κ 1,2 κ 2,1 κ 1 κ 2, I 22 = κ 1,22 + κ 1,1 κ 2,1 2κ2 2,1 κ 1 κ 2 + κ 1 κ 2 2 = κ 2,11 κ 1,2 κ 2,2 2κ2 1,2 κ 1 κ 2 + κ 2 1 κ 2, I 13 = κ 2,21 + 3κ 1,2 κ 2,1 κ 1 κ 2 = κ 2,12 κ 2,1 κ 2,2 2κ 1,2 κ 2,1 κ 1 κ 2, I 04 = κ 2,22 + 3κ2 2,1 κ 1 κ 2 +3κ 3 2. The two expressions for I 31 and I 13 follow from the commutator formula.

76 Third order recurrence relations: I 30 = D 1 κ 1 = κ 1,1, I 21 = D 2 κ 1 = κ 1,2, I 12 = D 1 κ 2 = κ 2,1, I 03 = D 2 κ 2 = κ 2,2, Fourth order recurrence relations: I 40 = κ 1,11 3κ2 1,2 κ 1 κ 2 +3κ 3 1, I 31 = κ 1,12 3κ 1,2 κ 2,1 κ 1 κ 2 = κ 1,21 + κ 1,1 κ 1,2 2κ 1,2 κ 2,1 κ 1 κ 2, I 22 = κ 1,22 + κ 1,1 κ 2,1 2κ2 2,1 κ 1 κ 2 + κ 1 κ 2 2 = κ 2,11 κ 1,2 κ 2,2 2κ2 1,2 κ 1 κ 2 + κ 2 1 κ 2, I 13 = κ 2,21 + 3κ 1,2 κ 2,1 κ 1 κ 2 = κ 2,12 κ 2,1 κ 2,2 2κ 1,2 κ 2,1 κ 1 κ 2, I 04 = κ 2,22 + 3κ2 2,1 κ 1 κ 2 +3κ 3 2. The two expressions for I 31 and I 13 follow from the commutator formula.

77 Fourth order recurrence relations I 40 = κ 1,11 3κ2 1,2 κ 1 κ 2 +3κ 3 1, I 31 = κ 1,12 3κ 1,2 κ 2,1 κ 1 κ 2 = κ 1,21 + κ 1,1 κ 1,2 2κ 1,2 κ 2,1 κ 1 κ 2, I 22 = κ 1,22 + κ 1,1 κ 2,1 2κ2 2,1 κ 1 κ 2 + κ 1 κ 2 2 = κ 2,11 κ 1,2 κ 2,2 2κ2 1,2 κ 1 κ 2 + κ 2 1 κ 2, I 13 = κ 2,21 + 3κ 1,2 κ 2,1 κ 1 κ 2 = κ 2,12 κ 2,1 κ 2,2 2κ 1,2 κ 2,1 κ 1 κ 2, I 04 = κ 2,22 + 3κ2 2,1 κ 1 κ 2 +3κ 3 2. The two expressions for I 22 imply the Codazzi syzygy κ 1,22 κ 2,11 + κ 1,1 κ 2,1 + κ 1,2 κ 2,2 2κ2 2,1 2κ2 1,2 κ 1 κ 2 κ 1 κ 2 (κ 1 κ 2 )=0, which can be written compactly as K = κ 1 κ 2 = (D 1 + Y 1 ) Y 1 (D 2 + Y 2 ) Y 2. = Gauss Theorema Egregium

78 Generating Differential Invariants From the general structure of the recurrence relations, one proves that the Euclidean differential invariant algebra I SE(3) is generated by the principal curvatures κ 1,κ 2 or, equivalently, the mean and Gauss curvatures, H, K, throughtheprocessofinvariantdifferentiation: I =Φ(H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... ) Remarkably, for suitably generic surfaces, the Gauss curvature can be written as a universal rational function of the mean curvature and its invariant derivatives of order 4: K =Ψ(H, D 1 H, D 2 H, D 2 1 H,..., D4 2 H) and hence I SE(3) is generated by mean curvature alone! To prove this, given K = κ 1 κ 2 = (D 1 + Y 1 ) Y 1 (D 2 + Y 2 ) Y 2 it suffices to write the commutator invariants Y 1,Y 2 in terms of H.

79 Generating Differential Invariants From the general structure of the recurrence relations, one proves that the Euclidean differential invariant algebra I SE(3) is generated by the principal curvatures κ 1,κ 2 or, equivalently, the mean and Gauss curvatures, H, K, throughtheprocessofinvariantdifferentiation: I =Φ(H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... ) Remarkably, for suitably generic surfaces, the Gauss curvature can be written as a universal rational function of the mean curvature and its invariant derivatives of order 4: K =Ψ(H, D 1 H, D 2 H, D 2 1 H,..., D4 2 H) and hence I SE(3) is generated by mean curvature alone! To prove this, given K = κ 1 κ 2 = (D 1 + Y 1 ) Y 1 (D 2 + Y 2 ) Y 2 it suffices to write the commutator invariants Y 1,Y 2 in terms of H.

80 Generating Differential Invariants From the general structure of the recurrence relations, one proves that the Euclidean differential invariant algebra I SE(3) is generated by the principal curvatures κ 1,κ 2 or, equivalently, the mean and Gauss curvatures, H, K, throughtheprocessofinvariantdifferentiation: I =Φ(H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... ) Remarkably, for suitably generic surfaces, the Gauss curvature can be written as a universal rational function of the mean curvature and its invariant derivatives of order 4: K =Ψ(H, D 1 H, D 2 H, D 2 1 H,..., D4 2 H) and hence I SE(3) is generated by mean curvature alone! To prove this, given K = κ 1 κ 2 = (D 1 + Y 1 ) Y 1 (D 2 + Y 2 ) Y 2 it suffices to write the commutator invariants Y 1, Y 2 in terms of H.

81 The Commutator Trick K = κ 1 κ 2 = (D 1 + Y 1 )Y 1 (D 2 + Y 2 )Y 2 To determine the commutator invariants: D 1 D 2 H D 2 D 1 H = Y 2 D 1 H Y 1 D 2 H D 1 D 2 D J H D 2 D 1 D J H = Y 2 D 1 D J H Y 1 D 2 D J H ( ) Non-degeneracy condition: ( D1 H D det 2 H D 1 D J H D 2 D J H ) 0, Solve ( ) fory 1, Y 2 in terms of derivatives of H, producingauniversalformula K =Ψ(H, D 1 H, D 2 H,... ) for the Gauss curvature as a rational function of the mean curvature and its invariant derivatives!

82 Definition. AsurfaceS R 3 is mean curvature degenerate if, near any non-umbilic point p 0 S, thereexistscalarfunctionsf 1 (t),f 2 (t) suchthat D 1 H = F 1 (H), D 2 H = F 2 (H). surfaces with symmetry: rotation, helical; minimal surfaces; constant mean curvature surfaces;??? Theorem. If a surface is mean curvature non-degenerate then the algebra of Euclidean differential invariants is generated entirely by the mean curvature and its successive invariant derivatives.

83 Minimal Generating Invariants A set of differential invariants is a generating system if all other differential invariants can be written in terms of them and their invariant derivatives. Euclidean curves C R 3 : Equi affine curves C R 3 : curvature κ and torsion τ affine curvature κ and torsion τ Euclidean surfaces S R 3 : mean curvature H Equi affine surfaces S R 3 : Pick invariant P. Conformal surfaces S R 3 : third order invariant J 3. Projective surfaces S R 3 : fourth order invariant K 4. = For any n 1, there exists a Lie group G N acting on surfaces S R 3 such that its differential invariant algebra requires n generating invariants! Finding a minimal generating set appears to be a very difficult problem. (No known bound on order of syzygies.)