---
title: Lie Curvatures in Differential and Lie Sphere Geometry
url: https://www.emergentmind.com/topics/lie-curvatures
type: topic
---

# Lie Curvatures in Differential and Lie Sphere Geometry

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Lie curvatures are Lie-invariant curvature quantities associated with objects studied in Lie sphere geometry and in differential geometry on Lie groups. Across the literature represented here, the expression does not denote a single universal scalar. Instead, it refers to several non-equivalent invariant constructions: cross-ratios of principal curvatures for proper Dupin hypersurfaces, Lie-invariant curvature densities and Euler–Lagrange equations for surfaces, four local invariants of generic transversal curves in Lie sphere geometry, harmonic curvature of curves in three-dimensional Lie groups, and sectional, Ricci, scalar, or Tanaka–Webster curvature quantities determined by Lie-algebraic or Lie-groupoid structure [2509.21235] [2310.15695] [2509.22408] [1211.6424] [2108.00651] [2606.29313].

## 1. Terminological scope

A common source of confusion is that “Lie curvature” is context-dependent. In the literature considered here, the phrase designates invariants attached to different geometric categories rather than a single canonical object. This suggests that the unifying feature is not a specific formula but invariance under a Lie-theoretic symmetry group: Lie sphere transformations, left- or right-invariant Lie-group symmetries, CR-contact symmetries, or Lie-groupoid/algebroid structures [2509.21235] [2606.04568].

| Setting | Geometric object | Lie curvature quantity |
|---|---|---|
| Proper Dupin hypersurfaces | Hypersurfaces in \(S^n\) | Cross-ratios of principal curvatures |
| Lie minimal surfaces | Umbilic-free surfaces in space forms | \( \frac{k_{1,u}k_{2,v}}{(k_1-k_2)^2} \) and its Euler–Lagrange PDEs |
| Generic transversal curves in \(\Lambda=T_1(S^3)\) | Curves in Lie sphere geometry | Four local invariants \(\kappa_1,\kappa_2,\kappa_3,\kappa_4\) |
| Curves in 3D Lie groups | Frenet curves with bi-invariant metric | Harmonic curvature \(H=\frac{\tau-T_G}{\kappa}\) |
| Riemannian/Lorentzian Lie groups | Left-invariant metrics | Sectional, Ricci, scalar curvature |
| Contact 3D Lie groups with CR structure | Embedded surfaces | Tanaka–Webster mean and Gauss curvature |

This multiplicity is mathematically significant. In Lie sphere geometry, Lie curvatures encode the projective geometry of curvature spheres. In Lie groups, they encode how curvature is algebraically determined by the bracket and an invariant metric. In pseudohermitian geometry on Lie groups, they encode how the contact, CR, and Reeb structures modify mean and Gauss curvature [2007.11947] [2104.11058] [1903.05194] [2209.02208].

## 2. Cross-ratio Lie curvatures of proper Dupin hypersurfaces

For a proper Dupin hypersurface \(M^{n-1}\subset S^n\) with distinct principal curvatures \(\kappa_1,\dots,\kappa_g\), a Lie curvature is the cross-ratio
\[
[\kappa_h,\kappa_i;\kappa_j,\kappa_k]
= \frac{(\kappa_h-\kappa_i)(\kappa_k-\kappa_j)}{(\kappa_h-\kappa_j)(\kappa_k-\kappa_i)}
\]
for four distinct indices \(h,i,j,k\) [2509.21235]. If one curvature is \(\infty\), as in the Legendre lift picture, the convention gives
\[
[a,b;c,\infty]=\frac{a-b}{a-c}.
\]
These quantities are invariant under Lie sphere transformations because they are projective invariants of the configuration of curvature spheres on a line in the Lie quadric [2509.21235].

When \(g=4\), and the principal curvatures are ordered by
\[
\kappa_1<\kappa_2<\kappa_3<\kappa_4,
\]
there is a unique Lie curvature
\[
\Lambda=[\kappa_1,\kappa_2;\kappa_3,\kappa_4], \qquad 0<\Lambda<1
\]
[2509.21235]. For isoparametric hypersurfaces with \(g=4\), Münzner’s result gives
\[
\Lambda\equiv \frac12,
\]
so constancy of Lie curvature is a rigidity signal in the proper Dupin problem [2509.21235].

The importance of this invariant became clear in the counterexamples to the Cecil–Ryan conjecture. In the Pinkall–Thorbergsson construction, the deformation \(V_{\alpha,\beta}\) of an FKM focal submanifold yields principal curvatures
\[
\kappa_1=-\frac{\alpha}{\beta},\qquad
\kappa_2=0,\qquad
\kappa_3=\frac{\beta}{\alpha},\qquad
\kappa_4=\infty,
\]
and hence
\[
\Lambda=\alpha^2
\]
at some points, while at the antipodal normal one obtains
\[
\Lambda=\beta^2
\]
[2509.21235]. Unless \(\alpha=\beta\), the Lie curvature is non-constant, so the example cannot be Lie equivalent to an isoparametric hypersurface. The Miyaoka–Ozawa Hopf-fibration construction produces compact proper Dupin hypersurfaces with \(g=4\) and \(g=6\) whose Lie curvatures are likewise non-constant [2509.21235]. These examples established that compact proper Dupin hypersurfaces need not be Lie equivalent to isoparametric hypersurfaces.

## 3. Lie-invariant surface curvatures in Lie sphere geometry

For surfaces, the phrase often shifts from cross-ratios to Lie-invariant curvature data attached to curvature spheres, gauge potentials, and curvature-line structures. In the Lie minimal surface theory of space forms, the fundamental Lie-invariant energy is
\[
L_{\text{Lie}}[X]
= \int_\Sigma \frac{k_{1,u}k_{2,v}}{(k_1-k_2)^2}\,du\wedge dv,
\]
written in curvature-line coordinates for an umbilic-free surface with principal curvatures \(k_1,k_2\) [2310.15695]. The density
\[
f=\frac{k_{1,u}k_{2,v}}{(k_1-k_2)^2}
\]
is Lie sphere invariant, so its critical points are Lie minimal surfaces [2310.15695].

The Euler–Lagrange equations are
\[
(k_2-k_1)k_{1,uv}+2k_{1,u}k_{1,v}=0,\qquad
(k_1-k_2)k_{2,uv}+2k_{2,u}k_{2,v}=0.
\]
These equations express Lie minimality purely in terms of principal curvatures and their first and second derivatives [2310.15695]. In this sense, the Lie curvature data of the surface are encoded by the energy density and the associated PDEs. The resulting rigidity is strong: constant mean curvature surfaces satisfying these Lie equations are rotational, and in \(\mathbb{R}^3\) the only non-planar minimal Lie minimal surface is the catenoid [2310.15695]. In Euclidean space, the same rotational conclusion holds for non-tubular affine Weingarten surfaces and for elliptic linear Weingarten surfaces with \(\Delta=b^2-ac>0\) [2310.15695].

A second surface-level invariant appears in the theory of Lie applicable surfaces. A Legendre map \(f:\Sigma\to\mathcal Z\) is Lie applicable if there exists
\[
\eta\in \Omega^1(f\wedge f^\perp)
\]
with
\[
d\eta=0,\qquad [\eta\wedge\eta]=0,
\]
such that the quadratic differential
\[
q(X,Y)=\operatorname{tr}\bigl(f\to f:\sigma\mapsto \eta(X)d_Y\sigma\bigr)
\]
is non-zero on a dense open set [2007.11947]. For a curved flat \(W\), this becomes
\[
q(X,Y)= -\frac{1}{2m}\operatorname{tr}\bigl(\mathcal N^W(X)\circ \mathcal N^W(Y)\big|_W\bigr),
\]
so \(q\) is the curvature metric shared by the associated Demoulin families [2007.11947]. This use of Lie curvature is not a scalar cross-ratio but a Lie-sphere-geometric quadratic differential controlling regularity, Darboux transformation, and curved-flat structure.

## 4. Lie curvatures of curves in Lie sphere geometry

For curves, Lie sphere geometry produces a Frenet-type theory on the unit tangent bundle
\[
\Lambda=T_1(S^3),
\]
viewed as a five-dimensional contact manifold acted on transitively by the Lie sphere group [2509.22408]. A generic transversal curve in \(\Lambda\) is an immersed curve everywhere transversal to the contact distribution. By the method of moving frames, such curves admit a Lie-invariant parameter, the Lie arclength \(s\), and are uniquely determined, up to Lie sphere transformation, by four local invariants
\[
\kappa_1,\kappa_2,\kappa_3,\kappa_4,
\]
called the Lie curvatures [2509.22408].

These four functions appear in the canonical Maurer–Cartan form of the normalized moving frame. The reconstruction theorem states that any smooth Lie arclength function together with any smooth \(\kappa_1,\kappa_2,\kappa_3,\kappa_4\) satisfying \(\kappa_1\neq 0\) determines a generic transversal curve uniquely up to the action of \(G\) [2509.22408]. This is the exact analogue, in Lie sphere geometry, of the Euclidean theorem that curvature and torsion determine a space curve up to rigid motion.

The simplest Lie-invariant functional is the Lie length
\[
\mathcal L[\gamma]=\int_\gamma ds.
\]
Its Euler–Lagrange equations imply that critical curves are characterized by constant Lie curvatures satisfying
\[
\kappa_1\neq 0,\qquad \kappa_3=0,\qquad \kappa_4=\kappa_1^2-\kappa_2^2
\]
[2509.22408]. The critical curves are therefore homogeneous: they arise as orbits of one-parameter subgroups of the Lie sphere group. In this setting, Lie curvatures are complete local invariants and simultaneously the natural variables of the variational problem [2509.22408].

## 5. Lie-adapted curvatures on Lie groups and CR manifolds

In the geometry of curves in a three-dimensional Lie group \(G\) with bi-invariant metric, the relevant Lie-adapted invariant is the harmonic curvature
\[
H=\frac{\tau-T_G}{\kappa},\qquad
T_G=\langle [T,N],B\rangle,
\]
where \(\{T,N,B,\kappa,\tau\}\) is the Frenet apparatus and \(T_G\) measures the Lie-bracket contribution to torsion [1211.6424]. In Euclidean \(\mathbb R^3\), \(T_G=0\), so \(H=\tau/\kappa\). In a 3D Lie group, however, \(H\) replaces \(\tau/\kappa\) in the classification of special curves: a general helix is characterized by \(H\) being constant, a Mannheim curve satisfies
\[
\lambda\,\kappa(1+H^2)=1,
\]
and a Bertrand curve satisfies
\[
\lambda\,\kappa+\mu\,\kappa H=1
\]
for constants \(\lambda,\mu\) [1211.6424]. Here “Lie curvature” means a curvature invariant corrected by the bracket structure of the ambient group.

A different Lie-theoretic usage concerns sectional, Ricci, and scalar curvature of Lie groups with invariant metrics. For \(GL(n,\mathbb R)\) and more general reductive Lie groups, sectional curvature can be written directly in terms of commutators and the Cartan decomposition. If
\[
u=u_1+u_2,\qquad v=v_1+v_2
\]
with \(u_1,v_1\in\mathfrak p\) and \(u_2,v_2\in\mathfrak k\), then
\[
\langle R(u,v)v,u\rangle
=
-2\|[u_1,v_1]\|^2
+\frac14\|[u,v]\|^2
+2\langle [u_1,v_1],[u_2,v_2]\rangle
\]
[2108.00651]. This formula makes curvature explicitly Lie-algebraic. In the tangent Lie group construction, even if the base group has constant negative sectional curvature, the tangent Lie group can exhibit positive, negative, and zero sectional curvatures, while in one family the Ricci curvature remains non-positive [1607.00367]. For three-dimensional unimodular and non-unimodular Lie groups with left-invariant Lorentzian metrics, the Ricci operator can have Segre types \(\{11,1\}\), \(\{1zz\}\), and \(\{21\}\), and the classification identifies flat metrics, constant-curvature metrics, semi-symmetric non locally symmetric metrics, and Ricci solitons [1903.05194] [2209.02208]. Another algebraic example is furnished by cyclic Riemannian Lie groups, defined by
\[
\oint_{X,Y,Z} h([X,Y],Z)=0,
\]
where the cyclic condition constrains the Levi–Civita connection and curvature [2504.16759].

On contact 3D Lie groups with a CR structure, the relevant invariants are Tanaka–Webster mean and Gauss curvature for embedded surfaces. If \(B\) denotes the Webster second fundamental form, then
\[
H_\Sigma^{TW}=-\frac{\mathrm{tr}(B)}{2},\qquad
K_\Sigma^{TW}=K(E_1,E_2)+\det B
\]
[2606.29313]. In the Heisenberg group, TW-cylindrical surfaces satisfy
\[
K_\Sigma^{TW}\equiv 0,
\]
whereas tilted planes are TW-minimal with positive TW Gauss curvature away from the characteristic point [2606.29313]. This is again a Lie curvature in the sense that the structure constants of the Lie algebra enter directly into the connection, torsion, and curvature formulas.

## 6. Integrability, rigidity, and unification

Several themes recur across these uses. The first is rigidity: cross-ratio Lie curvatures distinguish proper Dupin hypersurfaces from isoparametric ones, the Lie minimal PDEs force rotational symmetry under CMC or Weingarten constraints, and constant Lie curvatures characterize critical transversal curves in Lie sphere geometry [2509.21235] [2310.15695] [2509.22408].

The second is integrability. Lie applicable surfaces carry a pencil of flat connections \(d+t\eta\), quadratic differential \(q\), Demoulin families, and Darboux transforms [2007.11947]. Surfaces with spherical curvature lines are characterized by osculating sphere complexes \(L_i\) constant along the corresponding curvature directions, and a Lie applicable surface with exactly one family of spherical curvature lines must be generated by the lift of a constrained elastic curve in some space form [2104.11058]. Constrained elastic curves themselves admit a Lie sphere characterization via polynomial conserved quantities of a family of flat connections \(d+t\xi\) [2104.11058].

The third is unification at the level of Lie algebroids and Lie groupoids. Lie groupoids with right-invariant source metrics provide a common framework extending the Arnold–Milnor curvature formula for Lie groups and O’Neill’s formulas for Riemannian submersions [2606.04568]. In that framework, the sectional curvature of a source fibre is expressed by a groupoid version of the Arnold–Milnor “1–2–3–4” formula, and the Lie algebroid viewpoint subsumes Lie groups, principal bundles, and Riemannian submersion geometry within a single curvature formalism [2606.04568].

A plausible implication is that “Lie curvatures” are best understood not as one invariant but as a family of curvature constructions singled out by Lie-theoretic symmetry. In Lie sphere geometry they are projective invariants of curvature spheres and their deformations; in Lie groups they are curvature quantities whose formulas close algebraically on the bracket and invariant metric; in CR-contact settings they are pseudohermitian curvatures expressed by structure constants and horizontal derivatives. What unifies them is that curvature is encoded by the symmetry-adapted data of the relevant Lie structure rather than by arbitrary coordinates.

Source: https://www.emergentmind.com/topics/lie-curvatures