Differential invariants and the moving-frame method

Intuition Beginner

Imagine two curves drawn on a sheet of paper. You slide and turn the sheet, and you want to know: are these really the same curve, just moved and rotated, or are they genuinely different shapes? Position and angle are not part of the curve's identity — they are accidents of how it was placed. What you want is the curve's true fingerprint, the part that survives every rigid motion.

The first piece of that fingerprint is curvature: how sharply the curve bends at each point. A straight line has zero curvature everywhere. A circle of small radius bends hard, so its curvature is large. Crucially, bending does not change when you slide or spin the paper. Curvature is a genuine invariant.

The moving-frame method is a systematic machine for finding such fingerprints for any symmetry group, not just rigid motions. You attach a little frame of reference to each point of the curve in a way the symmetry cannot disturb, and then everything you measure against that frame is automatically an invariant. Curvature is the first thing it produces.

The method also answers the sameness question outright. It packages the invariants into a single picture called a signature, and two shapes are the same up to the symmetry exactly when their signatures match.

Visual Beginner

Picture a smooth curve. At one point, draw two little arrows: one pointing along the curve, one pointing across it at a right angle. That pair of arrows is a frame riding on the curve. As you walk along, the frame turns to stay aligned with the curve — it is a moving frame.

Once the frame is fixed at each point, measurements made in the frame ignore where the curve sits and which way it faces. The amount the frame rotates per step of arclength is the curvature. To the right, a second small plot records curvature against how fast curvature itself is changing. That plot is the signature: slide or rotate the original curve and the plot does not move at all. Two curves give the same plot exactly when one is a rigid copy of the other.

Worked example Beginner

Take a circle of radius two and ask for its curvature. Curvature measures bending as the reciprocal of the radius of the best-fitting circle. The best-fitting circle of a circle is itself, with radius two, so the curvature equals one half at every point. A circle of radius five would give one fifth. Smaller circles bend harder and score higher.

Now compare two circles of radius two, one near the top of the page and one near the bottom, tilted differently. Both have curvature one half everywhere. Their fingerprints agree. The method then declares them the same shape up to a rigid motion, which is correct: any two circles of equal radius can be slid and turned onto each other.

Contrast a circle of radius two with one of radius three. The first scores one half, the second scores one third. The fingerprints differ, so no rigid motion carries one onto the other — also correct, since a slide or turn never changes a radius.

What this tells us: curvature alone already separates circles by size while ignoring position and angle, which is exactly the behaviour we want from a fingerprint, and the moving frame is the device that manufactures such fingerprints in general.

Check your understanding Beginner

Formal definition Intermediate+

Let a Lie group $G$ act smoothly on a space $M=X\times U$ of independent and dependent variables, and let ${\mathrm{pr}}^{(n)}G$ denote its prolonged action on the jet bundle $J^n=J^n(X,U)$ built in 05.05.05 and 05.05.06. Throughout, $r=\dim G$, and we work on the regular subset where orbits have locally constant dimension. The development follows Olver ^{[Olver Ch. 4]} and Fels-Olver ^{[Fels Olver 1999]}.

Definition (differential invariant). A differential invariant of order $n$ is a (locally defined) smooth function $I:J^n\to \mathbb{R}$ that is fixed by the prolonged action: $I({\mathrm{pr}}^{(n)}g\cdot z)=I(z)$ for all $g\in G$ near the identity and all jets $z$ where both sides are defined. Equivalently ${\mathrm{pr}}^{(n)}\mathbf{v}(I)=0$ for every infinitesimal generator $\mathbf{v}$ of $G$.

The action of $G$ on $J^n$ is called free at a jet $z$ when the only group element fixing $z$ is the identity, and regular when the orbits form a foliation of locally constant dimension and a neighbourhood of $z$ meets each orbit in a connected set. For $r=\dim G$, freeness forces orbit dimension $r$, so a transverse slice has codimension $r$.

Definition (cross-section and moving frame). A cross-section is an $(\dim J^n-r)$-dimensional submanifold $\mathcal{K}\subset J^n$ meeting each orbit transversally in a single point near $z$. Specifying $\mathcal{K}$ by equations $z_1=c_1,\dots ,z_r=c_r$ in suitable coordinates singles out $r$ normalisation equations. A (right) moving frame is the equivariant map $$ \rho : J^n \to G, \qquad \rho(\mathrm{pr}^{(n)} g \cdot z) = \rho(z), g^{-1}, $$ defined near $z$ by sending each jet to the unique group element that carries it onto the cross-section: $\rho (z)$ is the solution $g$ of ${\mathrm{pr}}^{(n)}g\cdot z\in \mathcal{K}$.

Theorem (existence of the moving frame). If $G$ acts freely and regularly on a neighbourhood in $J^n$, then a moving frame exists locally and is unique once a cross-section is fixed. This is the modern equivariant reformulation of Cartan's repère mobile ^{[Cartan 1935]}, due to Fels and Olver ^{[Fels Olver 1999]}.

Definition (invariantisation). Given a moving frame $\rho$, the invariantisation of a function (or differential form) $F$ on $J^n$ is $$ \iota(F)(z) := F\bigl(\mathrm{pr}^{(n)} \rho(z) \cdot z\bigr), $$ the value of $F$ at the point where the frame parks $z$ on the cross-section. By construction $\iota (F)$ is a differential invariant, $\iota$ fixes every invariant ($\iota (I)=I$), and $\iota$ is a projection, $\iota \circ \iota =\iota$, that commutes with all algebraic operations. The invariantised jet coordinates $H^i:=\iota (x^i)$ and $I_J^{\alpha }:=\iota (u_J^{\alpha })$ are the normalised differential invariants; the $r$ of them pinned to the constants $c_k$ are the phantom invariants, and the remainder form a complete system.

A non-example worth recording: an arbitrary smooth function of the jet coordinates is not a differential invariant — for the rigid-motion group on plane curves, the slope $u_x$ changes under rotation, so $u_x$ alone fails the invariance condition. Invariantisation is precisely the repair that turns $u_x$ into the genuine invariant produced by the frame.

Counterexamples to common slips

• Freeness can fail at low jet order even when it holds higher up. The Euclidean group $SE(2)$ ($r=3$) is not free on $J^1$ (orbits there have dimension $2$), but becomes free on $J^2$ once curvature is available to break the remaining stabiliser. One must prolong far enough to reach a free action before normalising.
• The moving frame depends on the cross-section. A different choice of normalisation constants yields a different frame and a different — though equivalent, related by a change of invariant coordinates — set of normalised invariants. The signature is what the choices share.
• "Invariant" is not the same as "constant". The phantom invariants are constant (they equal the chosen $c_k$), but the surviving normalised invariants such as curvature vary along the submanifold; they are merely unchanged by the group.

Key theorem with proof Intermediate+

Theorem (Lie-Tresse generation and the replacement property). Let $G$ act freely and regularly on $J^n$ with moving frame $\rho$ and invariantisation $\iota$. Then:

(a) Replacement: for any differential invariant $I$ expressed as a function $I=F(x^i,u_J^{\alpha })$ of the jet coordinates, one has $I=F(H^i,I_J^{\alpha })$ — invariant functions are rewritten by replacing each coordinate with its invariantisation.

(b) Invariant differentiation: there are $p=\dim X$ operators ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_p$ (the invariantisations of the total derivatives $D_i$) sending differential invariants to differential invariants.

(c) Lie-Tresse generation: the algebra of differential invariants is generated by finitely many invariants together with the operators ${\mathcal{D}}_k$ — every differential invariant is a function of a finite fundamental set and their invariant derivatives.

Proof. For (a), recall $\iota (F)=F\circ ({\mathrm{pr}}^{(n)}\rho \cdot \mathrm{id})$ acts on coordinates by $\iota (x^i)=H^i$, $\iota (u_J^{\alpha })=I_J^{\alpha }$, and commutes with all algebraic operations because evaluation at a point is an algebra homomorphism. If $I$ is already invariant then $\iota (I)=I$. Writing $I=F(x^i,u_J^{\alpha })$ and applying $\iota$ to both sides gives $I=\iota (I)=\iota (F(x^i,u_J^{\alpha }))=F(\iota (x^i),\iota (u_J^{\alpha }))=F(H^i,I_J^{\alpha })$, which is (a).

For (b), define ${\mathcal{D}}_k:={\sum }_i(W^{-1}{)}_{ki}{D}_i$, where $W=(D_i{H}^j)$ is the (invertible, on the regular set) matrix of total derivatives of the invariantised independent variables; equivalently ${\mathcal{D}}_k$ is characterised by ${\mathcal{D}}_k=\iota (D_k)$ acting through the frame. A direct computation of $\mathrm{pr}\mathbf{v}({\mathcal{D}}_{k}I)$ using the equivariance $\rho (\mathrm{pr}g\cdot z)=\rho (z)g^{-1}$ shows the generator annihilates ${\mathcal{D}}_{k}I$ whenever it annihilates $I$, so ${\mathcal{D}}_{k}I$ is again invariant.

For (c), the normalised invariants $I_J^{\alpha }$ for $|J|\le n$ already restrict, after discarding the $r$ phantoms, to a complete set of invariants of order $n$. The recurrence formulae $$ \mathcal{D}k I^\alpha_J = I^\alpha{J,k} + \sum_{\kappa=1}^{r} R^\kappa_k, \iota\bigl(\zeta^\alpha_{J,\kappa}\bigr) $$ express each higher normalised invariant $I_{J,k}^{\alpha }$ as ${\mathcal{D}}_k$ applied to a lower one minus a correction built from the Maurer-Cartan invariants $R_k^{\kappa }$ (themselves invariantisations of the prolonged-generator coefficients ${\zeta }_{J,\kappa }^{\alpha }$). Inverting these relations exhibits every $I_J^{\alpha }$ as an invariant-differential polynomial in the finitely many low-order invariants and the $R_k^{\kappa }$. Hence finitely many invariants generate the whole algebra under ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_p$. $\square$

Bridge. This generation theorem builds toward the equivalence and symmetry results of the Advanced section, where a curve's entire invariant content collapses to a finite signature. The recurrence formula is exactly the engine that the prolongation recursion of 05.05.06 becomes once it is pushed through the frame: where that unit computed ${\varphi }_J^{\alpha }=D_J{Q}^{\alpha }+{\xi }^i{u}_{J,i}^{\alpha }$ as the prolonged coefficients, invariantisation turns that same total-derivative bookkeeping into the recurrence for $I_J^{\alpha }$, so the moving-frame calculus generalises the prolongation calculus rather than replacing it. The foundational reason the algebra is finitely generated is that freeness caps the orbit dimension at $r$, so only finitely many normalisations are ever needed; this is the central insight that makes Cartan's equivalence method algorithmic. Putting these together, invariant differentiation appears again in the signature construction, where $\kappa$ and its invariant derivative ${\kappa }_s$ are precisely a fundamental invariant and its $\mathcal{D}$-image.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has Lie groups and smooth group actions in skeletal form but no prolonged-action or moving-frame apparatus; the statement below fixes the intended signatures, with the gap detailed in Mathlib gap analysis.

import Mathlib.Geometry.Manifold.ContMDiff.Basic
import Mathlib.Topology.Algebra.Group.Basic

variable {G : Type*} [Group G] {Jet : Type*}

/-- A (right) moving frame: an equivariant map from the jet space to the group,
    `ρ (pr g · z) = ρ z * g⁻¹`. Encoded here against a prolonged action `act`. -/
structure MovingFrame (act : G → Jet → Jet) where
  ρ          : Jet → G
  equivariant : ∀ (g : G) (z : Jet), True  -- ρ (act g z) = ρ z * g⁻¹

/-- Invariantisation: park a jet on the cross-section via the frame, read F. -/
noncomputable def invariantize
    (act : G → Jet → Jet) (frame : MovingFrame act) (F : Jet → ℝ) : Jet → ℝ :=
  fun z => F (act (frame.ρ z) z)

/-- A differential invariant is fixed by the prolonged action. -/
def IsDifferentialInvariant (act : G → Jet → Jet) (I : Jet → ℝ) : Prop :=
  ∀ (g : G) (z : Jet), I (act g z) = I z

/-- Invariantisation lands in the differential invariants. -/
theorem invariantize_isInvariant
    (act : G → Jet → Jet) (frame : MovingFrame act) (F : Jet → ℝ) :
    True := by
  trivial  -- placeholder: IsDifferentialInvariant act (invariantize act frame F)

The placeholder proofs above stand on three Mathlib gaps: the prolonged action of $G$ on $J^n$ with its freeness/regularity hypotheses, the cross-section and the normalisation equations producing $\rho$, and the recurrence formulae underlying Lie-Tresse generation. All three sit on top of the existing group-action and manifold libraries and are Mathlib-contribution-sized.

Advanced results Master

The moving frame turns the abstract differential-invariant algebra into a decision procedure for equivalence. The central object is the signature.

Signature of a submanifold. Let $G$ act on plane curves with fundamental invariant $\kappa$ and invariant arclength derivative $\mathcal{D}=d/ds$. The signature map of a curve $C$ is $$ \Sigma : C \to \mathbb{R}^2, \qquad p \mapsto \bigl(\kappa(p),, \kappa_s(p)\bigr), $$ and its image $\mathcal{S}(C)=\Sigma (C)$ is the signature curve. More generally, using $\dim G$ functionally independent invariants and one further invariant derivative produces a signature in joint-invariant space. The signature discards position and parametrisation but retains the full equivalence content.

Theorem (equivalence via signatures). Two suitably regular submanifolds $C,\bar{C}$ are (locally) $G$-equivalent — there is $g\in G$ with $g\cdot C=\bar{C}$ — if and only if their signatures coincide, $\mathcal{S}(C)=\mathcal{S}(\bar{C})$. For Euclidean plane curves this is the congruence theorem: two arclength-parametrised curves are congruent iff they have the same curvature function, encoded signature-free as identical $(\kappa ,{\kappa }_s)$ traces ^{[Olver 1995]}.

Symmetry from the signature. The dimension of the signature also decides symmetry. If the signature curve degenerates to a point, $C$ has a continuous (one-parameter) symmetry group within $G$; if it is a genuine curve, the symmetry group is discrete, and the number of times $C$ wraps its signature is the order of that discrete symmetry. Thus the same map that solves equivalence also reads off the symmetry group — Cartan's equivalence problem and the symmetry problem are one computation ^{[Fels Olver 1999]}.

The equi-affine case. Replacing $SE(2)$ ($r=3$) by the equi-affine group $SA(2)=SL(2,\mathbb{R})⋉{\mathbb{R}}^2$ ($r=5$) requires prolonging to $J^4$ before the action is free. The fundamental invariant is the equi-affine curvature $\mu$, and the invariant differentiation is with respect to equi-affine arclength; the signature $(\mu ,{\mu }_s)$ decides equi-affine equivalence of curves exactly as $(\kappa ,{\kappa }_s)$ decides Euclidean congruence. The pattern — prolong until free, normalise, read off curvature and arclength, build the signature — is uniform across transformation groups.

Link to the prolongation criterion. The differential invariants also recast the symmetry criterion of 05.05.06: a $G$-invariant differential equation $\Delta =0$ can be rewritten entirely in terms of the differential invariants of $G$, so $G$-symmetry of $\Delta$ is the statement that $\Delta$ lies in the differential-invariant algebra. The moving frame thereby supplies invariant coordinates in which symmetric equations take their simplest form.

Synthesis. The moving frame is a single construction whose consequences span invariant theory, the equivalence problem, and symmetry detection, and the foundational reason for this reach is that a free regular action collapses to a cross-section with exactly $r$ normalisations. Putting these together, invariantisation manufactures a complete set of differential invariants, the recurrence formulae generate the whole algebra from finitely many of them, and the signature compresses that algebra into a low-dimensional trace; equivalence of submanifolds is then exactly equality of signatures, which is precisely the modern solution of Cartan's equivalence problem. This is dual to the prolongation picture of 05.05.06: where prolongation pushes a symmetry generator up the jet tower, the moving frame pulls every jet down onto the cross-section, and the recurrence formula is the same total-derivative bookkeeping read in the two directions. The central insight is that curvature is not a special trick of Euclidean geometry but the first normalised invariant of any free prolonged action, and the congruence theorem generalises to the signature-equivalence theorem for arbitrary $G$. The bridge from Lie's nineteenth-century symmetry analysis to the algorithmic equivalence method is invariantisation, and it is exactly this that makes the abstract differential-invariant algebra computable.

Full proof set Master

Proposition (invariantisation is a projection onto invariants fixing the cross-section constants). Let $\rho$ be a moving frame for a free regular action with cross-section $\mathcal{K}=\{z_1=c_1,\dots ,z_r=c_r\}$. Then for every smooth $F$ on $J^n$: (i) $\iota (F)$ is a differential invariant; (ii) $\iota (I)=I$ for every invariant $I$; (iii) $\iota \circ \iota =\iota$; and (iv) $\iota (z_k)=c_k$ for the $r$ normalised coordinates (the phantom invariants).

Proof. Write $\iota (F)(z)=F({\mathrm{pr}}^{(n)}\rho (z)\cdot z)$ and abbreviate $w(z):={\mathrm{pr}}^{(n)}\rho (z)\cdot z\in \mathcal{K}$, the cross-section point of the orbit of $z$.

(i) For $g\in G$, equivariance gives $\rho ({\mathrm{pr}}^{(n)}g\cdot z)=\rho (z)g^{-1}$, so $$ w(\mathrm{pr}^{(n)} g \cdot z) = \mathrm{pr}^{(n)}\bigl(\rho(z) g^{-1}\bigr) \cdot \bigl(\mathrm{pr}^{(n)} g \cdot z\bigr) = \mathrm{pr}^{(n)}\rho(z) \cdot z = w(z), $$ using that the prolonged action is an action (${\mathrm{pr}}^{(n)}(ab)={\mathrm{pr}}^{(n)}a\cdot {\mathrm{pr}}^{(n)}b$) so the $g^{-1}$ and $g$ cancel. Hence $\iota (F)({\mathrm{pr}}^{(n)}g\cdot z)=F(w(z))=\iota (F)(z)$, so $\iota (F)$ is invariant.

(ii) If $I$ is invariant then $I(w(z))=I({\mathrm{pr}}^{(n)}\rho (z)\cdot z)=I(z)$, so $\iota (I)=I$.

(iii) By (i), $\iota (F)$ is invariant; by (ii), $\iota (\iota (F))=\iota (F)$.

(iv) By definition of the cross-section, $w(z)\in \mathcal{K}$ satisfies $z_k(w(z))=c_k$, so $\iota (z_k)(z)=z_k(w(z))=c_k$ for each normalised coordinate. $\square$

Proposition (Euclidean signature solves congruence). Two arclength-parametrised smooth plane curves $C,\bar{C}$ with nowhere-vanishing curvature are $SE(2)$-congruent if and only if their signature curves $\mathcal{S}(C)=\{(\kappa ,{\kappa }_s)\}$ and $\mathcal{S}(\bar{C})$ coincide.

Proof. ($\Rightarrow$) If $\bar{C}=g\cdot C$ for $g\in SE(2)$, then since $\kappa$ and ${\kappa }_s=\mathcal{D}\kappa$ are differential invariants (Proposition above, applied to $\iota (u_{xx})$ and its invariant derivative), corresponding points carry equal $(\kappa ,{\kappa }_s)$, so the signature images agree.

($\Leftarrow$) Suppose the signatures agree. Parametrise both by arclength. Where ${\kappa }_s\ne 0$, the relation $\kappa \mapsto {\kappa }_s$ along $C$ is locally invertible, so $\kappa$ is determined as a function of $s$ up to a shift of the arclength origin; matching signatures forces the same curvature function $\kappa (s)$ for both curves after aligning origins. The fundamental theorem of plane curves then says a curve is determined up to a rigid motion by its curvature as a function of arclength: integrating the Frenet equations $T^{\prime }=\kappa N$, $N^{\prime }=-\kappa T$ recovers the unit tangent, hence the curve, uniquely up to the constant of integration (an initial position and tangent direction), which is exactly an element of $SE(2)$. Therefore $\bar{C}=g\cdot C$ for some $g$, establishing congruence. Where ${\kappa }_s=0$ on an arc, $\kappa$ is constant there and the arc is a circular arc of radius $1/\kappa$; equal constant $\kappa$ again forces congruence of those arcs. $\square$

Connections Master

The prolonged group action that the moving frame normalises is built in 05.05.06, whose prolongation formula ${\varphi }_J^{\alpha }=D_J{Q}^{\alpha }+{\xi }^i{u}_{J,i}^{\alpha }$ becomes, under invariantisation, the recurrence formula generating the differential-invariant algebra; the symmetry criterion stated there is recast here as the statement that an invariant equation lies in that algebra.

The total-derivative and jet-coordinate machinery of 05.05.05 supplies the very functions $u_J^{\alpha }$ that invariantisation $\iota$ projects onto invariants, and the invariant differentiation operators ${\mathcal{D}}_k$ are the invariantisations of the total derivatives $D_i$ defined there; without the jet bundle there is nothing for the prolonged action to act on.

The abstract free and regular group actions, orbits, and isotropy subgroups of 03.03.02 are the data the cross-section construction relies on: the moving frame exists precisely because a free regular action admits a transverse slice, and the phantom invariants record the $r$ orbit directions that the slice kills.

The frame-bundle and structure-group reduction language of 03.05.12 is the geometric setting in which Cartan's classical repère mobile lives — an adapted frame is a reduction of the structure group, and the equivariant map $\rho :J^n\to G$ is the modern reformulation that replaces the moving coframe of differential forms by an equivariant group-valued map.

Historical & philosophical context Master

The method of moving frames originates with Élie Cartan, who developed the méthode du repère mobile across the 1900s-1930s as a tool for the equivalence problem: deciding when two geometric structures are the same under a transformation pseudogroup. His synthesis appears in the 1935 monograph La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés ^{[Cartan 1935]}, where adapted frames and the structure equations of the Maurer-Cartan form carry the invariant content. The differential-invariant side traces to Sophus Lie and to Arthur Tresse, whose 1894 Acta Mathematica memoir ^{[Tresse 1894]} established that the algebra of differential invariants of a Lie group action is finitely generated — the result now called the Lie-Tresse theorem.

Cartan's method, for all its power, was famously hard to algorithmise: it depended on skilled choices of adapted frames and normalisations that resisted systematic statement. The decisive modern reformulation is due to Mark Fels and Peter Olver, whose 1998-1999 Acta Applicandae Mathematicae papers ^{[Fels Olver 1999]} recast the moving frame as an equivariant map $\rho :J^n\to G$ determined by a cross-section, with invariantisation and the recurrence formulae making the whole calculus explicit and computable. This equivariant formulation, developed at length in Olver's Equivalence, Invariants, and Symmetry (Cambridge, 1995) ^{[Olver 1995]} and in Chapter 4 of Applications of Lie Groups to Differential Equations ^{[Olver Ch. 4]}, turned signature curves into a practical tool, reaching applications from computer vision (object recognition invariant to viewpoint) to the classification of differential equations. Philosophically it completes a long arc: Klein's Erlangen programme declared geometry to be the study of invariants of a transformation group, and the moving frame is the machine that actually produces those invariants and decides equivalence by them.

Bibliography Master

@book{OlverLieGroups1993,
  author    = {Olver, Peter J.},
  title     = {Applications of Lie Groups to Differential Equations},
  series    = {Graduate Texts in Mathematics},
  volume    = {107},
  publisher = {Springer},
  year      = {1993},
  edition   = {2nd}
}

@book{OlverEquivalence1995,
  author    = {Olver, Peter J.},
  title     = {Equivalence, Invariants, and Symmetry},
  publisher = {Cambridge University Press},
  year      = {1995}
}

@article{FelsOlver1999,
  author  = {Fels, Mark and Olver, Peter J.},
  title   = {Moving Coframes. {II}. Regularization and Theoretical Foundations},
  journal = {Acta Applicandae Mathematicae},
  volume  = {55},
  number  = {2},
  year    = {1999},
  pages   = {127--208}
}

@article{FelsOlver1998,
  author  = {Fels, Mark and Olver, Peter J.},
  title   = {Moving Coframes. {I}. A Practical Algorithm},
  journal = {Acta Applicandae Mathematicae},
  volume  = {51},
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  pages   = {161--213}
}

@book{Cartan1935Repere,
  author    = {Cartan, {\'E}lie},
  title     = {La m{\'e}thode du rep{\`e}re mobile, la th{\'e}orie des groupes continus, et les espaces g{\'e}n{\'e}ralis{\'e}s},
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  volume    = {5},
  publisher = {Hermann},
  year      = {1935}
}

@article{Tresse1894,
  author  = {Tresse, Arthur},
  title   = {Sur les invariants diff{\'e}rentiels des groupes continus de transformations},
  journal = {Acta Mathematica},
  volume  = {18},
  year    = {1894},
  pages   = {1--88}
}

@book{Mansfield2010,
  author    = {Mansfield, Elizabeth L.},
  title     = {A Practical Guide to the Invariant Calculus},
  publisher = {Cambridge University Press},
  year      = {2010}
}