Lie Arclength in Lie Sphere Geometry

• Lie arclength is the invariant parameter in Lie sphere geometry, defined via a moving frame normalization on transversal curves.
• It uniquely classifies generic transversal curves by encoding four Lie curvatures (κ1, κ2, κ3, κ4) up to Lie sphere transformations.
• The concept extends invariant-length ideas to various geometries, analogous to affine arclength in equiaffine and related geometries.

Lie arclength is the Lie-invariant parameter arising in the Lie sphere geometry of generic curves in $\Lambda$, the unit tangent bundle of the unit $3$-sphere, when those curves are everywhere transversal to the contact distribution of $\Lambda$. By the method of moving frames, such curves can be parametrized by Lie arclength, and in that parametrization they are uniquely determined, up to Lie sphere transformation, by four local invariants called the Lie curvatures (Nicolodi, 26 Sep 2025). In a broader invariant-geometric sense, the literature also uses related arclength constructions associated with transformation groups, notably affine arc length in equiaffine geometry; this suggests that “Lie arclength” belongs to a family of group-invariant length elements rather than a single Euclidean-type metric notion (Clelland et al., 2012).

1. Lie sphere-geometric setting

The ambient model for the Lie sphere theory under discussion is

$\mathbb{R}^{4,2}=\mathbb{R}^6$

equipped with the bilinear form

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$

of signature $(4,2)$, where

$$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$

Let $G$ be the identity component of

$$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$

the Lie sphere group. The Lie quadric is

$$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$

The space $3$0 is the Grassmannian of null $3$1-planes through the origin in $3$2. Equivalently,

$3$3

and it is identified with the unit tangent bundle $3$4 of the unit $3$5-sphere. The chosen base point is

$3$6

and the projection

$3$7

is a principal $3$8-bundle (Nicolodi, 26 Sep 2025).

A fundamental feature of $3$9 is its contact structure. If $\Lambda$0 is the Maurer–Cartan form, then under any Lie frame field $\Lambda$1, the forms

$\Lambda$2

give a coframe on $\Lambda$3. Moreover,

$\Lambda$4

and therefore

$\Lambda$5

Thus $\Lambda$6 defines a contact structure on $\Lambda$7 (Nicolodi, 26 Sep 2025).

2. Transversal curves, nondegeneracy, and frame normalization

A smooth immersed curve

$\Lambda$8

is a transversal curve, or $\Lambda$9-curve, if it is everywhere transverse to the contact distribution, namely if

$\mathbb{R}^{4,2}=\mathbb{R}^6$0

for every local contact form $\mathbb{R}^{4,2}=\mathbb{R}^6$1. In the model $\mathbb{R}^{4,2}=\mathbb{R}^6$2, such a curve is an immersed curve $\mathbb{R}^{4,2}=\mathbb{R}^6$3 together with a unit vector field $\mathbb{R}^{4,2}=\mathbb{R}^6$4 along $\mathbb{R}^{4,2}=\mathbb{R}^6$5, whose tangential component never vanishes (Nicolodi, 26 Sep 2025).

A Lie frame field along $\mathbb{R}^{4,2}=\mathbb{R}^6$6 is a smooth lift

$\mathbb{R}^{4,2}=\mathbb{R}^6$7

such that $\mathbb{R}^{4,2}=\mathbb{R}^6$8. Because $\mathbb{R}^{4,2}=\mathbb{R}^6$9 is transversal, one can locally choose a first order Lie frame for which

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$0

For such a frame one has

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$1

Under change of first order frame,

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$2

so $$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$3 is invariant. The curve is nondegenerate if

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$4

everywhere. In that case one can normalize $$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$5, obtaining a second order frame (Nicolodi, 26 Sep 2025).

A further normalization yields a third order frame satisfying

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$6

Then

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$7

and $$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$8 is invariant. The curve is generic if

$$\langle x, y \rangle =-(x^0y^5 + x^5y^0)-(x^1y^4 + x^4y^1) +x^2y^2 +x^3y^3 = {}^t x\, h\, y,$$9

For a generic curve one can impose $(4,2)$0, thereby obtaining a fourth order frame. The remaining ambiguity is only $(4,2)$1, so the canonical frame lives in $(4,2)$2 (Nicolodi, 26 Sep 2025).

3. Definition of Lie arclength and Lie curvatures

For a second order frame, the $(4,2)$3-form $(4,2)$4 is frame-independent. Since

$(4,2)$5

there exists a strictly increasing function $(4,2)$6 such that

$(4,2)$7

This parameter $(4,2)$8 is the Lie arclength of the curve, uniquely determined up to an additive constant (Nicolodi, 26 Sep 2025).

In the generic case, the canonical frame $(4,2)$9 satisfies

$$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$0

where

$$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$1

The functions

$$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$2

are the Lie curvatures of $$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$3, and $$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$4. These four functions, together with the Lie arclength $$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$5, classify generic transversal curves up to Lie sphere transformation (Nicolodi, 26 Sep 2025).

The significance of Lie arclength is therefore structural rather than metric in the Euclidean sense. It is the parameter in which the canonical moving-frame equations assume their normalized form, and it is precisely the parameter with respect to which the four Lie curvatures become the complete local data recorded by the canonical frame (Nicolodi, 26 Sep 2025).

4. Existence, uniqueness, and reconstruction

The fundamental existence-and-uniqueness statement is the following: given a strictly increasing smooth function $$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$6 and smooth functions $$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$7 with $$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$8, there exists a generic transversal curve $$h= \begin{pmatrix} 0 & 0 & -L\ 0 & I_2 & 0\ -L & 0 & 0 \end{pmatrix}, \qquad L= \begin{pmatrix} 0&1\ 1&0 \end{pmatrix}, \qquad I_2= \begin{pmatrix} 1&0\ 0&1 \end{pmatrix}.$$9 having $G$0 as Lie arclength and $G$1 as Lie curvatures, unique up to Lie sphere transformation (Nicolodi, 26 Sep 2025).

This result places Lie arclength in the same role that Euclidean arc length occupies in classical Frenet theory, but within Lie sphere geometry. The curve is not described by Euclidean curvature and torsion; instead, after normalization by the moving frame, it is encoded by the Lie-invariant parameter $G$2 and four Lie curvatures. A plausible implication is that Lie arclength functions as the canonical independent variable for the local differential geometry of generic transversal curves in $G$3 (Nicolodi, 26 Sep 2025).

Because $G$4 is identified with $G$5, the reconstruction statement also links the invariant description to the geometry of contact elements on the $G$6-sphere. The data $G$7 determine the curve only up to the full Lie sphere group, not merely up to Euclidean rigid motion, which is exactly the natural equivalence in this geometry (Nicolodi, 26 Sep 2025).

5. The Lie-arclength functional and its Euler–Lagrange equations

The simplest Lie-invariant functional considered on generic transversal curves is

$G$8

The variational problem is posed on the space $G$9 of generic $$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$0-curves parametrized by Lie arclength, with compactly supported variations through generic $$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$1-curves (Nicolodi, 26 Sep 2025).

The prolonged frame data are encoded in

$$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$2

On $$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$3, the Pfaffian system $$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$4 is generated by

$$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$5

with independence condition

$$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$6

The integral curves of $$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$7 are exactly the prolonged frames of generic transversal curves, and the derived flag has constant rank: $$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$8 The analysis uses Griffiths’ exterior differential systems approach to the calculus of variations (Nicolodi, 26 Sep 2025).

The criticality criterion is explicit. A generic $$\{A\in GL(6,\mathbb{R})\mid {}^tAhA=h\}\cong O(4,2),$$9-curve $$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$0, parametrized by Lie arclength, is a critical curve of $$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$1 if and only if its Lie curvatures are constant and satisfy

$$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$2

Equivalently,

$$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$3

If $$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$4 is critical, then the canonical frame satisfies

$$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$5

with

$$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$6

Hence

$$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$7

for some fixed $$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$8, and

$$\mathcal Q=\{[x]\in \mathbb{RP}^5\mid \langle x,x\rangle=0\}.$$9

Thus the critical curves of the Lie-arclength functional are exactly the orbits of one-parameter subgroups of the Lie sphere group through the base point $3$00 (Nicolodi, 26 Sep 2025).

6. Broader invariant-arclength context and terminological scope

In the broader theory of invariant curve geometry, the phrase “Lie arclengths” refers to invariant length elements associated with transformation groups acting on curves and surfaces. In equiaffine $3$01-space $3$02, for example, a nondegenerate curve has affine arc length

$3$03

while a nondegenerate surface carries the affine first fundamental form

$3$04

and a curve in that surface acquires the induced affine length

$3$05

These two affine-invariant arc length notions generally differ. The paper introducing this comparison states that it fits into the general theory of Lie arclengths and identifies exactly when the two arc lengths agree (Clelland et al., 2012).

A separate, but related, usage appears in harmonic analysis, where affine arclength measure is the natural replacement for ordinary arclength when studying Fourier restriction to possibly degenerate curves in $3$06. For a $3$07 curve $3$08,

$3$09

In that context, the phrase “Lie/affine arclength” refers to a canonical geometric measure on curves that generalizes ordinary arclength in a way compatible with affine harmonic analysis (Bak et al., 2011).

The terminology is not uniform across all uses of “arclength” near Lie-theoretic problems. In the study of line tracking in spherical images, the relevant quantity is the sub-Riemannian arclength on the Lie group $3$10, contrasted with spherical arclength on $3$11; there the group parameter measures length with respect to a left-invariant sub-Riemannian structure (Mashtakov et al., 2016). By contrast, in pseudo-arclength continuation for structural avalanches in amorphous carbon, the arclength is explicitly a numerical continuation parameterization and “is not referring to ‘Lie arclength’ or any differential-geometric path length associated with Lie groups or flows” (Birks et al., 30 Jan 2026).

Within this terminological landscape, Lie arclength in the strict sense is the invariant parameter produced by moving-frame normalization in a geometry governed by a transformation group. In the Lie sphere setting, that parameter is $3$12 defined by

$3$13

and its role is inseparable from the four Lie curvatures and from the classification and variational theory of generic transversal curves in $3$14 (Nicolodi, 26 Sep 2025).