Inductive Approach to Cartan s Moving Frame Method. Irina Kogan Yale University

Transcription

1 Inductive Approach to Cartan s Moving Frame Method Irina Kogan Yale University July, 2002

2 Generalized definition of a moving frame M. Fels and P.J. Olver (1999) A moving frame is an equivariant smooth map ρ: M G. G R g 1 G ρ ρ M g Theorem. A moving frame exists if the action is regular and free. M

3 A (local) moving frame free action and (local) cross-section K: codim K = dim O z, K is transversal to O z for z U M, K O z consists of at most one point. Define ρ(z), by the condition ρ(z) z K ρ(g z)g z = ρ(z) z freeness ρ(g z) = ρ(z)g 1 Invariantization of functions f : M R ι(f)(z) = f(ρ(z) z) Z i = ι(z i ) a complete set of functionally independent invariants.

4 Euclidean Geometry on the plane. The Frénet Frame. SE(2) = SO(2) R 2 Infinitesimal arc-length ds = ) 1 + u 2 x dx ( dx T = ds, du dt, ds ds = κn Basic differential invariants: the Euclidean curvature κ and its derivatives κ s = dκ ds, κ ss, etc. Affine Geometry on the plane. SA(2) = SL(2) R 2 Infinitesimal arc-length dα = uxx 1/3 dx ( dx T = dα, du ), N = dt = T N = 1 dα dα dn dα = µt Basic differential invariants: affine curvature µ and its derivatives µ α = dµ dα, µ αα, etc.

5 Moving Frames on Homogeneous Spaces. P.A. Griffiths (1974), M.L.Green (1978) N Euclidean Example: f = f f 1 u x 1+u 2 x 1+u 2 x 1+u u x 1 2 x 1+u 2 x ( dt ds, dn ds, dx ) ds G G/H x u f 1 d ds ( f) ds = = (T, N, X) = (T, N, X). 0 κ 1 κ κds ds κds ,

6 Jet spaces: J k = J k (M, p) are bundles over M. The fiber over z M consists of the equivalence classes of p-dim. submanifolds with k-th order contact at z. Local coordinates on J k : {x 1,..., x p, u 1,... u q, u α J }, where J = (j 1,..., j p ) J = j j p k, j i 0. Projections: J J k J k 1... J 1 J 0 = M π n k : Jt J k for t k Jets of submanifolds: If N : u α = f α (x) then j k (N) : u α = f α (x), u α J = k f α Prolongation of the action: g j k (N) = j k (g N). j 1x 1... j px p.

7 Theorem. (Ovsyannikov, Olver) The action of G is loc. effective on open subsets n dim G (the order of stabilization), s. t. the prolonged action is loc. free on an open dense V n J n. moving frame ρ : V n G ρ can be lifted to J k for k > n: ρ(z (k) ) = ρ ( π k n(z (k) ) ) complete set of differential invariants, invariant differential forms, invariant differential operators.

8 Euclidean action on the plane: x u y = cos(φ)x sin(φ)u + a v = sin(φ)x + cos(φ)u + b u x v 1 = sin(φ) + cos(φ)u x cos(φ) sin(φ)u x u xx u xx v 2 = (cos(φ) sin(φ)u x ) 3 u xxx v 3 = (cos(φ) sin(φ)u x)u xxx + 3sin(φ)u 2 xx (cos(φ) sin(φ)u x ) 5 cross-section on J 1 : K E = {x = 0, u = 0, u x = 0} moving frame is the solution of y = 0, v = 0, v 1 = 0 : φ = arctan(u x ), Substitute in v 2 and v 3 : I e 2 = κ = a = x + u xu, u 2 x + 1 u xx (1 + u 2 x )3/2 b = u xx u u 2 x + 1 I e 3 = κ s = (1 + u2 x )u xxx 3u x u 2 xx (1 + u 2 x )3

9 dy = cos(φ)dx sin(φ)du = (cos(φ) sin(φ)u x ) dx sin(φ)θ = d H y + d V y, where θ = du u x dx d H y = (cos(φ) sin(φ)u x ) dx, d V y = sin(φ)θ moving frame φ = arctan(u x ) d H y ds = 1 + u 2 x dx

10 Modifications: Recursive algorithm (some conditions on the group action) Construct differential invariants on J k order by order, at each step normalizing more and more of the group parameters at the end obtaining a moving frame for the group G. The structure of invariant differential forms on J k. Inductive algorithm G = AB s. t. A B is discrete. invariants and moving frames for A and B can be used to construct invariants and a moving frame for G. Relations among the invariants of G and its subgroups.

11 Inductive algorithm: G = BA, B A is discrete Proposition 1. n a and a local c.-s. K A J n a s. t. A acts loc. freely near K A, K A is invariant under the action of B. m.f. ρ A : J n a A defined by ρ A (z (n a) ) z (n a ) KA. for k n a the action of B is well defined on: K k A = {z(k) J k π k n a (z (k) ) K a }.

12 contains a complete set of G- invariants. Proposition 2. n s. t. B acts loc. freely on K n A = {z(k) J n π n n a (z (k) ) K A }. c.-s. K K n a for B-action. m.f. ρ B : K n A B defined by ρ B(z (n) ) z (n) K Proposition 3. K is a c.-s. for the G-action. ρ G : J n G is a m.f. for G. ρ G (z (n) ) = ρ B (ρ A (z (n) ) z (n) )ρ A (z (n) ) ρ B (ρ A (z ( ) ) z ( ) )ρ A (z ( ) ) z ( )

13 From the Euclidean to the affine action: SA(2, R) = SL(2, R) R 2 = B SE(2, R) {( )} τ λ B = 0 1 τ where B is the isotropy group of z (1) 0 = {x = 0, u = 0, u x = 0} = K E. dim SA(2, R) = 5 lowest inv. on J 4. Step 1: Prolong the action of B to J 4 : x τx + λu, u 1 τ u, u x u xx u x τ(τ + λu x ), u xx (τ + λu x ) 3, u xxx (τ + λu x)u xxx 3λu 2 xx (τ + λu x ) 5, u xxxx (τ + λu x) 2 u xxxx 10(τ + λu x )λu xx u xxx + 15λ 2 u 3 xx (τ + λu x ) 7.

14 Step 2 Restrict these transformations to K 4 E = {z(4) π 4 1 (z(4) ) = z (1) 0 } = {z (4) x = 0, u = 0, u x = 0} : u xx u xx τ 3, u xxx τu xxx 3λu 2 xx τ 5, u xxxx τ2 u xxxx 10τλu xx u xxx + 15λ 2 u 3 xx τ 7. Step 3 Choose a cross-section for B action K (4) = {z (4) K 4 E u xx = 1, u xxx = 0} M. f. ρ B : K 4 E B: τ = (u xx ) 1/3 and λ = u xxx 3(u xx ) 5/3. invariant for B action on K 4 E : I b 4 = u xxu xxxx 5 3 (u xxx) 2 (u xx ) 8/3.

15 Step 4 Substitute Euclidean invariants: In terms of κ : µ = I a 4 = Ie 2 Ie (Ie 3 )2 (I e 2 )8/3 I e 2 = κ, Ie 3 = κ s, I e 4 = κ ss + 3κ 3 µ = κ(κ ss+3κ 3 ) 5 3 κ2 s κ 8/3. Differentiation with respect to the affine arc length: dα = κ 1/3 ds d dα = 1 κ 1/3 d ds

16 Affine invariant differential operator: x y = τx + λu d H y = (τ + λu x ) dx B-moving frame: τ = (u xx ) 1/3 and λ = u xxx 3(u xx ) 5/3. ( u 1/3 xx + ) u xxx u x 3(u xx ) 5/3 dx. Invariantization with respect to the Euclidean action. Note that on K 4 E : u x = 0, u xx = I e 2 = κ, u xxx = I e 3 = κ s, dx = ds dα = κ 1/3 ds d dα = 1 κ 1/3 d ds

17 From the affine to the projective action: PSL(3, R) acts on R 2 : x αx + βu + γ δx + ɛu + ζ ; λx + νu + τ u δx + ɛu + ζ PSL(3, R) = B SA(2, R) B = is the isotropy group of 1 ab 0 0 a 0 b c 1 a K A = z (3) 0 = {x = 0, u = 0, u 1 = 0, u 2 = 1, u 3 = 0} K 7 A = {z(7) π 7 3 (z(7) ) = z (3) 0 } restrict the prolonged B-action to K 7 A. choose cross-section for B-action to K 7 A : K = {z (7) K 7 A u 4 = 0, u 5 = 1, u 6 = 0} normalize parameters a, b and c and substitute affine invariants:

18 Projective curvature in terms of affine invariants: η = 7µ2 αα + 6µ αµ ααα 3µµ 2 α 6µ 8/3 α Invariant differential operator for the projective action in terms of the affine: d dϱ = 1 d µ 1/3 α dα

19 The moving frame for the projective group is the product of the matrices: µ αα 3 µ α 0 0 µ 1/3 µ αα µ α 4/ α 0 µ 2 αα 3µµ2 α µ 7/3 α 1 µ 1/3 α κ 1/3 1 κ s 3 κ 5/ κ 1/ u u x 1+u 2 2 u x x x 1 1+u 2 x 1+u 2 x uu x+x 1+u 2 x xu x u 1+u 2 x

20 Affine and Euclidean action for curves in R 3 SA(3, R) = B SE(3, R) and B SA(3, R) = I, where B = is the isotropy group of a b c 0 f g 0 0 af 1. K E = {z (2) x = 0, u = 0, v = 0, u x = 0, v x = 0, v xx = 0} I u,a 5 = 1 36 κ 4 τ 7/2 ( 36κ 2 τ 2 (τκ sss κ ss τ s + 4κ 2 τκ s κτ 2 τ s 3κ 3 τ s + 2τ 3 κ s )+ 60τ 2 κ(κ 2 sτ s 3τκ ss κ) 6κ 2 τ(τ 2 s κ s 3κτ ss τ s )+ ) 160κ 3 s τ3 25κ 3 τs 3, I v,a 5 = 1 24 κ 8/3 τ 7/3 ( 36κ 2 τ 2 (κ 2 + τ 2 ) 20κ 2 sτ 2 8κτκ s τ s 35κ 2 τ 2 s + 12κτ(τκ ss + 2κτ ss ) ). dα = (κ 2 τ) 1/6 ds

21 Recursive algorithm: G acts on M regularly but not freely a cross-section K 0. ρ 0 : M G: ρ 0 (z) z K 0. is defined up to the isotropy groups on K. Coordinates of ρ 0 (z) z a complete set of 0-th order invariants.

22 Example: G = SO(3, R) acting on R 3. K 0 is the upper half z-axis 0-th order invariant: r = x 2 + y 2 + z 2 Each point on K 0 has the same isotropy group H SO(2, R) [ρ 0 ] : M H\G

23 Definition: Isotypic slice is a c.-s. with the same isotropy group at each point. isotypic slice all orbits are of the same type + proper action (e.g. G is compact). Algorithm (0) K 0 M with isotropy group H 0 G [ρ 0 ] : M H 0 \G is G-equivariant. H 0 acts on K0 1 = {z(1) π0 1(z(1) ) K 0 }. (1) K 1 K 1 0 with isotropy group H 1 H 0 [τ 1 ] : K 1 0 H 1\H 0 is H 0 -equivariant. [ρ 1 ](z (1) ) = [ τ 1 (ρ 0 (z (1) ) z (1) )ρ 0 (z (1) ) ] : J 1 H 1 \G is G-equivariant. Coordinates of ρ 1 (z (1) )z (1) a complete set of 1-st order invariants.

24 J n ([ρ n ], ι n ) (H n \G) K n π n n 1 (δ n n 1, πn n 1 ).. π 2 1 π 1 0 J 2 ([ρ 2 ], ι 2 ) (H 2 \G) K 2 J 1 ([ρ 1 ], ι 1 ) (H 1 \G) K 1 M (H 0 \G) K 0 ([ρ 0 ], ι 0 ) (δ 2 1, π2 1 ) (δ 1 0, π1 0 ) H k 1 H k and π k k 1 (K k) = K k 1

25 G-equivariant maps: π k k 1 jet projections Jk+1 J k, ρ k : J k H k \G, δ k k 1 ([g] k) = [g] k 1 : H k \G H k 1 \G. where [g] k is the image of g under G H k \G. G-invariant map: ι k (z (k) ) = ρ k (z (k) ) z (k) : J k K k.