Élie cartan's theory of moving framesornarnalds/files/malstofaglaerur.pdf · 2017. 12. 4. ·...

Élie Cartan’s Theory of Moving Frames

’Orn Arnaldsson

Department of MathematicsUniversity of Minnesota, Minneapolis

Special Topics Seminar, Spring 2014

The Story of Symmetry

Felix Klein and Sophus Lie (1870s) develop a theory of“continuous transformation groups”.Klein publishes his famous Erlangen program in 1872,characterizing all known non-Euclidean geometries as thestudy of various homogeneous spaces (a notion based onthe continuous group concept.)Lie utilizes the infinitesimal part of the new theory (themodern Lie algebras) in a search for a “Galois Theory” ofdifferential equations.His work centers around finding invariants of group actionsand symmetry of the equations.Later (1900s), Émile Picard and Ernest Vessiot completeLie’s vision for ODEs.Lie and Klein did not have global notions such asmanifolds.

In the hands of Élie Cartan (1910-1930) the 19th centuryfinite dimensional transformation groups became themodern Lie groups, the objects that permeate so much ofmathematics and physics.Also, Cartan created a technique to study geometricobjects, “repère mobile” or moving frames.Unfortunately, reading Cartan’s papers is notoriouslydifficult so the technique was not developed further whilethe mathematics community struggled understandingCartan.Finally, S.S. Chern (1940s), a collaborator of Cartan’s,popularized the method among geometers.Still, there was never a rigorous mathematical formulationof the method, and it was only known to work in specialcases, until Olver and Fels provided it in 1997.

Hermann Weyl

Cartan’s theory of moving frames was based on his remarkablegeometrical insight, rather than clear, rigorous formulations.

“I did not quite understand how he [Cartan] does this in general,though in the examples he gives the procedure is clear.“

“Nevertheless, I must admit I found the book, like most ofCartan’s papers, hard reading.”

- Hermann Weyl

“Cartan on groups and differential geometry”Bull. Amer. Math. Soc. 44 (1938) 598-601

Hermann Weyl

Cartan’s theory of moving frames was based on his remarkablegeometrical insight, rather than clear, rigorous formulations.

“I did not quite understand how he [Cartan] does this in general,though in the examples he gives the procedure is clear.“

“Nevertheless, I must admit I found the book, like most ofCartan’s papers, hard reading.”

- Hermann Weyl

“Cartan on groups and differential geometry”Bull. Amer. Math. Soc. 44 (1938) 598-601

The finite and the infinite

Consider the rotation group SO(2) = S1 acting on theplane R2:

(x ,u)→ (x cos(θ)− u sin(θ), x sin(θ) + u cos(θ)).

Consider transformations on M = (x ,u) ∈ R2 | u 6= 0,defined by

(x ,u)→ (f (x),u

f ′(x)),

for arbitrary local diffeomorphisms f . This example of aninfinite dimensional pseudo-group of transformations wasknown to Sophus Lie. The prefix “pseudo” stems from thefact that composition of two such transformations may notbe possible (the image of one must lie in the domain of theother.)

The finite and the infinite

Consider the rotation group SO(2) = S1 acting on theplane R2:

(x ,u)→ (x cos(θ)− u sin(θ), x sin(θ) + u cos(θ)).

Consider transformations on M = (x ,u) ∈ R2 | u 6= 0,defined by

(x ,u)→ (f (x),u

f ′(x)),

for arbitrary local diffeomorphisms f . This example of aninfinite dimensional pseudo-group of transformations wasknown to Sophus Lie. The prefix “pseudo” stems from thefact that composition of two such transformations may notbe possible (the image of one must lie in the domain of theother.)

The objects of study

A smooth (right) action of a Lie group G on a manifold M is asmooth map Ψ : M ×G→ M such that

Ψ(x ,gh) = Ψ(Ψ(x ,g),h), ∀ x ∈ M, g,h ∈ G.

Some remarks:Whenever convenient, we shall simply write x · g instead ofΨ(x ,g). The above condition then readsx · (gh) = (x · g) · h.For a fixed g ∈ G, the map x → x · g has inversex → x · g−1.The set Ox := x · g | g ∈ G is called the orbit of Gthrough x ∈ M.A smooth map f : M → N is called invariant under theaction if f (x · g) = f (x), ∀ x ∈ M, g ∈ G. This is equivalentto f being constant on the orbits of G in M.

Ψ(x ,gh) = Ψ(Ψ(x ,g),h), ∀ x ∈ M, g,h ∈ G.

Origin of the moving frame

Cartan developed his theory of moving frames for Lie groupactions in an attempt to solve the congruence problem: Giventwo submanifolds N, N ⊆ M, determine whether or not there isa group element, g, such that N = N · g.

The well known Frénet-Serret frame is the classical example ofa moving frame. In this case, the group action is translationsand rotations of curves in R3. Two curves are congruent underthe group action if and only if their Frénet-Serret frames areequivalent.

The key element to the early results on moving frames was thenaturality of the exterior derivative

ϕ∗d = dϕ∗.

The modern definition, due to Olver and Fels (1997), takes adifferent route.

ϕ∗d = dϕ∗.

Jets and jet bundles

The most difficult conceptual construction in the theory ofsymmetry is the jet bundle, a fundamental concept in thedifferential geometric theory of partial differential equations.

We shall describe the local coordinate structure of jet bundles.This means considering a manifold of the typeM = X × U ⊆ Rp+q, where X ⊆ Rp is interpreted as the spaceof independent variables and U ⊆ Rq the space of dependentvariables.

Jets and jet bundles

The most difficult conceptual construction in the theory ofsymmetry is the jet bundle, a fundamental concept in thedifferential geometric theory of partial differential equations.

We shall describe the local coordinate structure of jet bundles.This means considering a manifold of the typeM = X × U ⊆ Rp+q, where X ⊆ Rp is interpreted as the spaceof independent variables and U ⊆ Rq the space of dependentvariables.

The simplest case

For convenience, let p = q = 1. The first jet space, J(1)(M) isthe space M × R = X × U × R, where the third coordinate is tobe interpreted as the first derivative, and a general point inJ(1)(M) is written (x ,u,ux ).

Analogously, the k th jet space is J(k)(M) = M × Rk and ageneral point has the form (x ,u,ux , . . . ,ukx ), where weinterpret ukx as being dk

dxk u.

If G acts smoothly on M = X × U, there is an induced action ofG on J(1)(M), called the prolonged action, described as follows.

The simplest case

dxk u.

The simplest case

dxk u.

Prolonged action

Consider (x0,u0,ux0) ∈ J(1)(M) and let f : X → U be a smoothfunction (possibly just defined on a neighborhood of x0) whosegraph, Γf ⊆ M, “passes through” the point (x0,u0,ux0). That is,f (x0) = u0 and f ′(x0) = ux0.

For g ∈ G close enough to the identity, the curve Γf · g will bethe graph of a smooth function, f , on some neighborhoodaround x0, where (xo, u0) := (x0,u0) · g. Define

ux0 :=d

∣∣∣∣x=x0

f (x).

We define the prolonged action of G on J(1)(M) according to

(x0,u0,ux0) · g := (x0, u0, ux0).

Prolonged action

Consider (x0,u0,ux0) ∈ J(1)(M) and let f : X → U be a smoothfunction (possibly just defined on a neighborhood of x0) whosegraph, Γf ⊆ M, “passes through” the point (x0,u0,ux0). That is,f (x0) = u0 and f ′(x0) = ux0.

For g ∈ G close enough to the identity, the curve Γf · g will bethe graph of a smooth function, f , on some neighborhoodaround x0, where (xo, u0) := (x0,u0) · g. Define

ux0 :=d

∣∣∣∣x=x0

f (x).

We define the prolonged action of G on J(1)(M) according to

(x0,u0,ux0) · g := (x0, u0, ux0).

Prolongation

•(x0, u0)

ux0•

(x0, u0)

This definition of prolonged action does not depend on thechoice of function, f , “representing” the point (x0,u0,ux0) in jetspace.

Prolongation

•(x0, u0)

ux0•

(x0, u0)

This definition of prolonged action does not depend on thechoice of function, f , “representing” the point (x0,u0,ux0) in jetspace.

Prolongation of SO(2)

The prolonged action of SO(2) on R2 by rotations to the firstand second order jet spaces is given by

(x ,u)→ (x cos(θ)− u sin(θ), x sin(θ) + u cos(θ)),

ux →sin(θ) + ux cos(θ)

cos(θ)− ux sin(θ),

uxx →uxx

(cos(θ)− ux sin(θ))3 .

Equi-affine geometry of curves in R2

This geometry is governed by the action of the special affinegroup SA(2) = SL(2) nR2 on R2 via

(x ,u)→ (δ(x−a)−β(u−b),−γ(x−a)+α(u−b)), αδ−βγ = 1.

The first few prolongations of this action are

ux → −γ − αux

δ − βux, uxx → −

(δ − βux )3 ,

uxxx → −(δ − βux )uxxx + 3βu2

xx(δ − βux )5 ,

uxxxx → −(δ − βux )2uxxxx + 10β(δ − βux )uxxuxxx + 15β2u3

xx(α + βux )7 .

Equi-affine geometry of curves in R2

This geometry is governed by the action of the special affinegroup SA(2) = SL(2) nR2 on R2 via

(δ − βux )3 ,

xx(δ − βux )5 ,

xx(α + βux )7 .

Lie’s “Galois Theory”

Let ∆(x ,u(n)) = 0, x ∈ X ⊆ Rp, u ∈ U ⊆ Rq, be a differentialequation of order n. A group, G, acting on the spaceM = X × U, is said to be a symmetry of ∆(x ,u(n)) = 0 if itmoves graphs of (local) solutions to ∆ = 0 to other (local)solutions.

Given the symmetry group of ∆ = 0, it is sometimes possible toreduce the order of ∆ = 0. In favorable cases, a symmetrygroup of dimension r allows the order to be reduced by r .

When the order can be reduced to zero, in the classicalterminology, ∆ = 0 is said to have been solved by quadratures.The term Galois Theory comes from Lie’s attempt to classifythose differential equations that can be solved in this way, inanalogy to solvability of polynomials.

The infinitesimal

Lie’s great discovery was of infinitesimal methods for findingsymmetry groups of differential equations.

His theorem says that a function f : M → N is invariant underthe action of G on M (f (x · g) = f (x)) if and only if v(f ) = 0 forall elements v ∈ g, in the Lie algebra of G, where v(f ) is theinduced action of g on M.

The major advantage of this result is not having to deal with thecomplicated non-linear condition f (x · g) = f (x) and insteaddeal with the linear condition v(f ) = 0.

The infinitesimal

Reducing order

Consider the homogeneous ODE

ux = F (x/u).

This equation has symmetry group R+ acting on X ×U = R2 by

(x ,u) · λ→ (xλ,uλ).

A process suggested by Lie, of “straightening out” thesymmetry, gives rise to a change of variables

y = x/u, w = log(x),

but in these variables the original equation becomes

F (y)− y,

an equation of order zero in wy , which can be integrated(solved by quadratures.)

Reducing order

ux = F (x/u).

(x ,u) · λ→ (xλ,uλ).

F (y)− y,

Reducing order

ux = F (x/u).

(x ,u) · λ→ (xλ,uλ).

F (y)− y,

N’other’s Theorem

In the case of systems of Euler-Lagrange equations comingfrom a variational problem with functional

∫Ω L(x ,u(n))dx , Lie’s

infinitesimal theory leads to the celebrated N’other Theorem, afundamental result in the calculus of variations and physics.

The theorem says that there is a one-to-one correspondencebetween one dimensional symmetry groups of the functionaland conservation laws of the system.

The calculus of variations has been geometrically formulatedinto the Variational Bi-complex where the moving frame theoryhas found important applications.

Freeness and Regularity

The ultimate definition of a moving frame for Lie group actionsrequires us to restrict to group actions that are free and regular.

An action Ψ as above is free if x · g = x for some x ∈ M, g ∈ G,means that g = e, where e is the identity element of G.

Regularity of an action means roughly that the group orbitscannot behave too wildly. Regularity guarantees that the set ofgroup orbits has a manifold structure.

Prolonging until free

When confronted with an action that isn’t free we prolong theaction to a suitably high order jet bundle hoping it will eventuallybecome free.

A result by Ovsiannikov says that for locally effective actions,the prolonged action will eventually become locally free (on aneighborhood of the identity.)

For locally free actions, by the inverse function theorem, thegroup orbits have the same dimension as the Lie group. Inparticular, we can have no hope of freeness if the manifold hasstrictly smaller dimension than the Lie group.

Freeness of SA(2) acting on R2

In this case, we have a five dimensional Lie group, SA(2),acting on a two dimensional manifold R2.

For there to be any hope of freeness, we will at least have toprolong this action to the five dimensional space J(3)(R2) = R5.We already demonstrated the prolonged action on the thirdorder jet bundle.

To check for local freeness, Olver has given a condition basedon the Lie determinant, similar to the Wronskian for lineardependence of functions. The action of SA(2) turns out to befree on J(3).

Freeness of SA(2) acting on R2

In this case, we have a five dimensional Lie group, SA(2),acting on a two dimensional manifold R2.

For there to be any hope of freeness, we will at least have toprolong this action to the five dimensional space J(3)(R2) = R5.We already demonstrated the prolonged action on the thirdorder jet bundle.

To check for local freeness, Olver has given a condition basedon the Lie determinant, similar to the Wronskian for lineardependence of functions. The action of SA(2) turns out to befree on J(3).

The Moving Frame

Here is the definition of moving frame, due to Olver and Fels.

DefinitionA (left) moving frame for a free and regular action of a Lie groupG, on a manifold M, is a (left) equivariant map ρ : M → G.

Left equivariance means that ρ(x · g) = g−1 · ρ(x).

Do not be alarmed if you fail to see the connection of thisdefinition with geometrical objects such as the Frénet-Serretframe.

The Moving Frame

Here is the definition of moving frame, due to Olver and Fels.

DefinitionA (left) moving frame for a free and regular action of a Lie groupG, on a manifold M, is a (left) equivariant map ρ : M → G.

Left equivariance means that ρ(x · g) = g−1 · ρ(x).

Do not be alarmed if you fail to see the connection of thisdefinition with geometrical objects such as the Frénet-Serretframe.

The Fundamental Theorem

The conditions of freeness and exactness of the group actionare actually the best we can do.

Fundamental Theorem on Moving Frames

A moving frame exists for a smooth action of a Lie group on amanifold if and only if the action is free and regular.

The Fundamental Theorem

The conditions of freeness and exactness of the group actionare actually the best we can do.

Fundamental Theorem on Moving Frames

A moving frame exists for a smooth action of a Lie group on amanifold if and only if the action is free and regular.

Invariants

For a general group action, it can be very difficult finding theinvariant functions satisfying f (x · g) = f (x). Even using Lie’ssimplifying linear condition, v(f ) = 0, finding f requires usingthe method of characteristics to solve a first order PDE.

However, a moving frame makes this task trivial!

Indeed, the function I : M → M, taking x → x · ρ(x) gives acomplete list of invariants of the action, in that any otherinvariant is a function of I. The powerful Frobenius theoremfrom differential geometry guarantees this.

To see that I is invariant, simply note that

I(x · g) = (x · g) · ρ(x · g) = x · g · g−1 · ρ(x) = x · ρ(x) = I(x).

Invariants

Constructing a moving frame

The construction of a moving frame for a free and regular groupaction on an m dimensional manifold with r ≤ m dimensionalorbits, depends on a choice of a local cross-section to thegroup orbits.

A local cross section to the orbits means an m − r dimensionalsubmanifold, κ, that intersects each orbit exactly once in someneighborhood in the manifold.

We can then define a moving frame ρ(x) ∈ G to be the unique(by freeness) element of G that moves x onto the cross section,

x · ρ(x) ∈ κ.

Cross section and a moving frame

•xρ(x)

Equivariance

It is easy to see that this procedure gives a local equivariantmap ρ : M → G. Indeed, the unique element of G that movesx · g into κ is g−1 · ρ(x),

x · g · g−1 · ρ(x) = x · ρ(x) ∈ κ,

by the definition of ρ(x).

So ρ(x · g) = g−1 · ρ(x).

Equivariance

It is easy to see that this procedure gives a local equivariantmap ρ : M → G. Indeed, the unique element of G that movesx · g into κ is g−1 · ρ(x),

x · g · g−1 · ρ(x) = x · ρ(x) ∈ κ,

by the definition of ρ(x).

So ρ(x · g) = g−1 · ρ(x).

Invariantization

For any smooth function, F : M → N, defined on the domain ofour moving frame, we can invariantize it, defining “a version” ofF , ι(F ), such that ι(F )(x) is the value of F at the cross section,where the orbit through x intersects it.

This means defining the invariantization of F to be the function

ι(F )(x) := F (x · ρ(x)).

Notice that we first calculate F (x · g) and then plug in theformula for the moving frame, i.e. putting g = ρ(x).

This invariatization process can be generalized to othergeometrical objects, where we essentially pull back the objectof interest to the cross section, thereby making them invariant.

Invariantization

ι(F )(x) := F (x · ρ(x)).

Invariantization

ι(F )(x) := F (x · ρ(x)).

Invariantization

ι(F )(x) := F (x · ρ(x)).

Invariantizing in the cotangent bundle

On the domain of definition of the moving frame mapρ : M → G, we can invariantize differential forms on M.Skipping the theoretical details, invariantizing the localcoordinate one forms dx i is particularly easy in practise.

We simply calculate the differential of the action on M, given byx → x · g, and then replace g by ρ(x) in the outcome.

Invariantizing any smooth basis for the cotangent bundle, weobtain a G-invariant coframe, $1, . . . , $m.

It is the dual frame of this coframe that gives classical movingframe vectors.

Example: SO(2)nR2 acting on R2

Expanding on our running example of SO(2), the semi directproduct SO(2) nR2 = SE(2) acts on R2 by rotations andtranslations,

(x ,u)→ (x cos(θ)− u sin(θ) + a, x sin(θ) + u cos(θ) + b),

where a,b ∈ R.Within the Erlangen program, thishomogeneous space represents “high-school geometry”.

SE(2) is a three dimensional Lie group and prolonging to thefirst jet space of R2, J(1)(R2) = R3 its action becomes free.

In J(1)(R2), with coordinates (x ,u,ux ), the action in the thirdvariable is

ux →ux cos(θ)− sin(θ)

cos(θ)− ux sin(θ).

Continued

The group orbits in J(1)(R2) are three dimensional, so anycross section has to be zero dimensional. Let the pointκ = x = u = ux = 0 be our cross section.

To construct a moving frame for this cross section, for eachpoint (x ,u,ux ) ∈ J(1)(R2), we need to determine a g ∈ SE(2)such that (x ,u,ux ) · g ∈ κ. This means solving the followingequations for θ,a and b.

0 = x cos(θ)− u sin(θ) + a,

0 = x sin(θ) + u cos(θ) + b,

0 =ux cos(θ)− sin(θ)

Continued

The group orbits in J(1)(R2) are three dimensional, so anycross section has to be zero dimensional. Let the pointκ = x = u = ux = 0 be our cross section.

To construct a moving frame for this cross section, for eachpoint (x ,u,ux ) ∈ J(1)(R2), we need to determine a g ∈ SE(2)such that (x ,u,ux ) · g ∈ κ. This means solving the followingequations for θ,a and b.

0 = x cos(θ)− u sin(θ) + a,

0 = x sin(θ) + u cos(θ) + b,

0 =ux cos(θ)− sin(θ)

A moving frame

The solution to this system of equations is

θ = − tan−1(ux ) a = x , b = u,

and so a moving frame ρ : J(1) → SE(2) is given by

(x ,u,ux )→ (−θ,a,b) = (tan−1(ux ), x ,u).

Invariantizing

Prolonging to J(2), we have an action on the extra variable uxx ,

uxx →uxx

The cross section κ as a subset in J(2) is now a onedimensional submanifold (the uxx axis.)

The three dimensional group SE(2) is acting freely on the fourdimensional space J(2) = R4 and the Frobenius theorem tellsus that there is exactly 4-3=1 invariant function found in thecomponents of z → z · ρ(z), where z = (x ,u,ux ,uxx ).

Indeed, using the moving frame ρ, constructed above, we findthat this function sends (x ,u,ux ,uxx ) to

(0,0,0,uxx

(1 + u2x )3/2

Invariantizing

uxx →uxx

(0,0,0,uxx

(1 + u2x )3/2

Invariantizing

uxx →uxx

(0,0,0,uxx

(1 + u2x )3/2

Invariantizing

uxx →uxx

(0,0,0,uxx

(1 + u2x )3/2

Curvature

We see that at the second order (on J(2)) there is just onefundamental invariant,

κ =uxx

(1 + u2x )3/2

which is the familiar curvature from two dimensional Euclideangeometry.

This process can now be repeated, prolonging to ever higherdegree and calculating the new invariant at every step.

However, this is not the best way to obtain all differentialinvariants, as we shall see.

Curvature

κ =uxx

(1 + u2x )3/2

Curvature

κ =uxx

(1 + u2x )3/2

Frénet-Serret in R2

Let us see how this modern formulation provides us with theclassical moving frame in R2.

Calculating the differential of the R2 → R2 map x → x · g gives

cos(θ)dx − sin(θ)du

sin(θ)dx + cos(θ)du.

Plugging the moving frame formulaθ = − tan−1(ux ), a = x , b = u into these equations gives thefollowing coframe on R2

1√1 + u2

x(dx + uxdu)

1√1 + u2

x(−uxdx + du).

1√1 + u2

x(dx + uxdu)

1√1 + u2

x(−uxdx + du).

1√1 + u2

x(dx + uxdu)

1√1 + u2

x(−uxdx + du).

The dual vectors to this coframe form the classical movingframe

1√1 + u2

∂x+ ux

∂u),

1√1 + u2

x(−ux

∂u).

Equi-affine geometry

As another example of a moving frame, consider the action ofthe special affine group SA(2) = SL(2) nR2 on R2 via

(δ − βux )3 ,

xx(δ − βux )5 ,

xx(α + βux )7 .

Equi-affine geometry

As another example of a moving frame, consider the action ofthe special affine group SA(2) = SL(2) nR2 on R2 via

(δ − βux )3 ,

xx(δ − βux )5 ,

xx(α + βux )7 .

Normalizing the parameters

This action is free on J(3) = R5. Letκ = x = u = ux = 0,uxx = 1,uxxx = 0 and solve thefollowing equations for the parameters α, β, γ, δ, a, b,

0 = δ(x − a)− β(u − b), 0 = −γ(x − a) + α(u − b)

0 = −γ − αux

δ − βux, 1 = − uxx

(δ − βux )3 ,

0 = −(δ − βux )uxxx + 3βu2xx

(δ − βux )5 .

Moving frame

The solution to the above system is

α = 3√

uxx , β = −13

u−5/3xx uxxx ,

γ = ux3√

uxx , δ = u−1/3xx − 1

3u−5/3

xx uxxx ,

a = x , b = u.

Equi-affine curvature

Invariantizing the fourth jet coordinate

xx(α + βux )7 ,

by plugging into its formula the newly found moving frame,gives a the differential invariant

κ =3uxxuxxxx − 5u2

3u8/3xxx

This is the well known equi-affine curvature.

Equi-affine curvature

Invariantizing the fourth jet coordinate

xx(α + βux )7 ,

by plugging into its formula the newly found moving frame,gives a the differential invariant

κ =3uxxuxxxx − 5u2

3u8/3xxx

This is the well known equi-affine curvature.

Invariant operators

Using the remarkable theory of contact forms, which is afundamental ingredient in the geometric theory of PDEs, wecan find so-called invariant differential operators that generatethe full algebra of differential invariants through repeatedlyacting on finitely many fundamental invariants.

A differential operator, D, is called invariant, if D(K ) is invariantfor any invariant K .

In practice, such operators are found by invariantizing thehorizontal coframe on the space of independent variables (theX in M = X × U) and taking its dual frame.

Invariant operators

Arc-length

Going back to the action of SE(2) on R2 there is only the onehorizontal form dx .

Invariantizing it gives a one form ι(dx) = ds =√

1 + u2x dx . This

is the familiar arc-length element in Euclidean geometry.

Differentiating with respect to the dual of the arc-length elementcorresponds to differentiating with respect to arc length. Thedifferential operator is given by

D =1√

1 + u2x

Now, every differential invariant can be obtained by repeatedapplication of the invariant D to the curvature κ. For example,the third order invariant is

D(κ) =dκds

=uxxx (1 + u2

x )3/2 − 32(1 + u2

x )1/2 · 2uxu2xx

(1 + u2x )7

Arc-length

1 + u2x dx . This

D =1√

1 + u2x

D(κ) =dκds

=uxxx (1 + u2

x )3/2 − 32(1 + u2

x )1/2 · 2uxu2xx

(1 + u2x )7

Arc-length

1 + u2x dx . This

D =1√

1 + u2x

D(κ) =dκds

=uxxx (1 + u2

x )3/2 − 32(1 + u2

x )1/2 · 2uxu2xx

(1 + u2x )7

Arc-length

1 + u2x dx . This

D =1√

1 + u2x

D(κ) =dκds

=uxxx (1 + u2

x )3/2 − 32(1 + u2

x )1/2 · 2uxu2xx

(1 + u2x )7

The differential invariant algebra

The collection of differential invariants and the invariantdifferential operators form a so-called D-module, and theiralgebraic structure is of great interest.

The Fundamental Basis Theorem says that for any free andregular action on M, the entire differential algebra of invariantscan be generated by a finite set of fundamental invariantsthrough repeated invariant differentiation. This assumes therole of The Hilbert Basis Theorem for polynomial ideals.

An open problem is the determination of a minimal generatingset of differential invariants. There is some hope that Gr’obnerbases techniques in conjunction with results on syzygyclassification based on the moving frame, can give insight intothis problem.

QUESTIONS?

Élie cartan's theory of moving framesornarnalds/files/malstofaglaerur.pdf · 2017. 12. 4. ·...

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