Lie Curvatures in Differential and Lie Sphere Geometry

• Lie curvatures are Lie-invariant measures defined by diverse geometric constructions, capturing key characteristics of surfaces, curves, and Lie groups.
• They are applied via cross-ratio invariants for Dupin hypersurfaces, Lie minimal surface PDEs, and Lie-adapted curvature concepts in CR and group settings.
• Their study uses moving frame techniques, curvature energy functionals, and Lie-algebroid frameworks to reveal rigidity, integrability, and unification in geometry.

Searching arXiv for the supplied papers and closely related terminology to ground the article in current literature. arxiv_search.query({"^^^^^^^^18search_query18 minimal Weingarten surfaces\"^^^^^^^^18 OR id:(Cho et al., 2023)18^^^^^^^^"," arxiv_search.query({"^^^^^^^^18search_query18 a variational problem for curves in Lie sphere geometry\"^^^^^^^^18ti:\18search_query18 arxiv_search.query({"^^^^^^^^18search_query18 curvatures\"","^^^^^^^^18max_results18 arxiv_search.query({"^^^^^^^^18search_query18 of Compact Dupin Hypersurfaces with Non-constant Lie Curvatures\" OR id:(&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&)","^^^^^^^^18max_results18 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 (&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&, &&&^^^^18search_query18^^^^&&&, &&&^^^^18ti:\18^^^^&&&, &&&^^^^18sort_order18^^^^&&&, &&&^^^^18descending18^^^^&&&, &&&^^^^18search_query18^^^^&&&).

^^^^18ti:\18^^^^. 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 (&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&, &&&^^^^18ti:\18search_query18^^^^&&&).

Setting	Geometric object	Lie curvature quantity
Proper Dupin hypersurfaces	Hypersurfaces in PRESERVED_PLACEHOLDER_^^^^18search_query18^^^^	Cross-ratios of principal curvatures
Lie minimal surfaces	Umbilic-free surfaces in space forms	PRESERVED_PLACEHOLDER_^^^^18ti:\18^^^^ and its Euler–Lagrange PDEs
Generic transversal curves in PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18^^^^	Curves in Lie sphere geometry	Four local invariants PRESERVED_PLACEHOLDER_^^^^18max_results18^^^^
Curves in ^^^^18max_results18^^^^D Lie groups	Frenet curves with bi-invariant metric	Harmonic curvature PRESERVED_PLACEHOLDER_^^^^18sort_by18^^^^
Riemannian/Lorentzian Lie groups	Left-invariant metrics	Sectional, Ricci, scalar curvature
Contact ^^^^18max_results18^^^^D 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 (&&&^^^^18ti:\18ti:\18^^^^&&&, &&&^^^^18ti:\18 OR id:(Cho et al., 2023)18^^^^&&&, &&&^^^^18ti:\18max_results18^^^^&&&, &&&^^^^18ti:\18sort_by18^^^^&&&).

^^^^18 OR id:(Cho et al., 2023)18^^^^. Cross-ratio Lie curvatures of proper Dupin hypersurfaces

For a proper Dupin hypersurface PRESERVED_PLACEHOLDER_^^^^18submittedDate18^^^^ with distinct principal curvatures PRESERVED_PLACEHOLDER_^^^^18sort_order18^^^^, a Lie curvature is the cross-ratio

PRESERVED_PLACEHOLDER_^^^^18descending18^^^^

for four distinct indices PRESERVED_PLACEHOLDER_^^^^18search_query18^^^^ (&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&). If one curvature is PRESERVED_PLACEHOLDER_^^^^18ti:\18^^^^, as in the Legendre lift picture, the convention gives

PRESERVED_PLACEHOLDER_^^^^18ti:\18search_query18^^^^

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 (&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&).

When PRESERVED_PLACEHOLDER_^^^^18ti:\18ti:\18^^^^, and the principal curvatures are ordered by

PRESERVED_PLACEHOLDER_^^^^18ti:\18 OR id:(Cho et al., 2023)18^^^^

there is a unique Lie curvature

PRESERVED_PLACEHOLDER_^^^^18ti:\18max_results18^^^^

(&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&). For isoparametric hypersurfaces with PRESERVED_PLACEHOLDER_^^^^18ti:\18sort_by18^^^^, Münzner’s result gives

PRESERVED_PLACEHOLDER_^^^^18ti:\18submittedDate18^^^^

so constancy of Lie curvature is a rigidity signal in the proper Dupin problem (&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&).

The importance of this invariant became clear in the counterexamples to the Cecil–Ryan conjecture. In the Pinkall–Thorbergsson construction, the deformation PRESERVED_PLACEHOLDER_^^^^18ti:\18sort_order18^^^^ of an FKM focal submanifold yields principal curvatures

PRESERVED_PLACEHOLDER_^^^^18ti:\18descending18^^^^

and hence

PRESERVED_PLACEHOLDER_^^^^18ti:\18search_query18^^^^

at some points, while at the antipodal normal one obtains

PRESERVED_PLACEHOLDER_^^^^18ti:\18ti:\18^^^^

(&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&). Unless PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18search_query18^^^^, 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 PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18ti:\18^^^^ and PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18 OR id:(Cho et al., 2023)18^^^^ whose Lie curvatures are likewise non-constant (&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&). These examples established that compact proper Dupin hypersurfaces need not be Lie equivalent to isoparametric hypersurfaces.

^^^^18max_results18^^^^. 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

PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18max_results18^^^^

written in curvature-line coordinates for an umbilic-free surface with principal curvatures PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18sort_by18^^^^ (&&&^^^^18search_query18^^^^&&&). The density

PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18submittedDate18^^^^

is Lie sphere invariant, so its critical points are Lie minimal surfaces (&&&^^^^18search_query18^^^^&&&).

The Euler–Lagrange equations are

PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18sort_order18^^^^

These equations express Lie minimality purely in terms of principal curvatures and their first and second derivatives (&&&^^^^18search_query18^^^^&&&). 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 PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18descending18^^^^ the only non-planar minimal Lie minimal surface is the catenoid (&&&^^^^18search_query18^^^^&&&). In Euclidean space, the same rotational conclusion holds for non-tubular affine Weingarten surfaces and for elliptic linear Weingarten surfaces with PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18search_query18^^^^ (&&&^^^^18search_query18^^^^&&&).

A second surface-level invariant appears in the theory of Lie applicable surfaces. A Legendre map PRESERVED_PLACEHOLDER_^^^^18 OR id:(Cho et al., 2023)18ti:\18^^^^ is Lie applicable if there exists

PRESERVED_PLACEHOLDER_^^^^18max_results18search_query18^^^^

with

PRESERVED_PLACEHOLDER_^^^^18max_results18ti:\18^^^^

such that the quadratic differential

PRESERVED_PLACEHOLDER_^^^^18max_results18 OR id:(Cho et al., 2023)18^^^^

is non-zero on a dense open set (&&&^^^^18ti:\18ti:\18^^^^&&&). For a curved flat PRESERVED_PLACEHOLDER_^^^^18max_results18max_results18^^^^, this becomes

PRESERVED_PLACEHOLDER_^^^^18max_results18sort_by18^^^^

so PRESERVED_PLACEHOLDER_^^^^18max_results18submittedDate18^^^^ is the curvature metric shared by the associated Demoulin families (&&&^^^^18ti:\18ti:\18^^^^&&&). 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.

^^^^18sort_by18^^^^. Lie curvatures of curves in Lie sphere geometry

For curves, Lie sphere geometry produces a Frenet-type theory on the unit tangent bundle

PRESERVED_PLACEHOLDER_^^^^18max_results18sort_order18^^^^

viewed as a five-dimensional contact manifold acted on transitively by the Lie sphere group (&&&^^^^18ti:\18^^^^&&&). A generic transversal curve in PRESERVED_PLACEHOLDER_^^^^18max_results18descending18^^^^ 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 PRESERVED_PLACEHOLDER_^^^^18max_results18search_query18^^^^, and are uniquely determined, up to Lie sphere transformation, by four local invariants

PRESERVED_PLACEHOLDER_^^^^18max_results18ti:\18^^^^

called the Lie curvatures (&&&^^^^18ti:\18^^^^&&&).

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 PRESERVED_PLACEHOLDER_^^^^18sort_by18search_query18^^^^ satisfying PRESERVED_PLACEHOLDER_^^^^18sort_by18ti:\18^^^^ determines a generic transversal curve uniquely up to the action of PRESERVED_PLACEHOLDER_^^^^18sort_by18 OR id:(Cho et al., 2023)18^^^^ (&&&^^^^18ti:\18^^^^&&&). 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

PRESERVED_PLACEHOLDER_^^^^18sort_by18max_results18^^^^

Its Euler–Lagrange equations imply that critical curves are characterized by constant Lie curvatures satisfying

PRESERVED_PLACEHOLDER_^^^^18sort_by18sort_by18^^^^

(&&&^^^^18ti:\18^^^^&&&). 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 (&&&^^^^18ti:\18^^^^&&&).

^^^^18submittedDate18^^^^. Lie-adapted curvatures on Lie groups and CR manifolds

In the geometry of curves in a three-dimensional Lie group PRESERVED_PLACEHOLDER_^^^^18sort_by18submittedDate18^^^^ with bi-invariant metric, the relevant Lie-adapted invariant is the harmonic curvature

PRESERVED_PLACEHOLDER_^^^^18sort_by18sort_order18^^^^

where PRESERVED_PLACEHOLDER_^^^^18sort_by18descending18^^^^ is the Frenet apparatus and PRESERVED_PLACEHOLDER_^^^^18sort_by18search_query18^^^^ measures the Lie-bracket contribution to torsion (&&&^^^^18sort_order18^^^^&&&). In Euclidean PRESERVED_PLACEHOLDER_^^^^18sort_by18ti:\18^^^^, PRESERVED_PLACEHOLDER_^^^^18submittedDate18search_query18^^^^, so PRESERVED_PLACEHOLDER_^^^^18submittedDate18ti:\18^^^^. In a ^^^^18max_results18^^^^D Lie group, however, PRESERVED_PLACEHOLDER_^^^^18submittedDate18 OR id:(Cho et al., 2023)18^^^^ replaces PRESERVED_PLACEHOLDER_^^^^18submittedDate18max_results18^^^^ in the classification of special curves: a general helix is characterized by PRESERVED_PLACEHOLDER_^^^^18submittedDate18sort_by18^^^^ being constant, a Mannheim curve satisfies

PRESERVED_PLACEHOLDER_^^^^18submittedDate18submittedDate18^^^^

and a Bertrand curve satisfies

PRESERVED_PLACEHOLDER_^^^^18submittedDate18sort_order18^^^^

for constants PRESERVED_PLACEHOLDER_^^^^18submittedDate18descending18^^^^ (&&&^^^^18sort_order18^^^^&&&). 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 PRESERVED_PLACEHOLDER_^^^^18submittedDate18search_query18^^^^ and more general reductive Lie groups, sectional curvature can be written directly in terms of commutators and the Cartan decomposition. If

PRESERVED_PLACEHOLDER_^^^^18submittedDate18ti:\18^^^^

with PRESERVED_PLACEHOLDER_^^^^18sort_order18search_query18^^^^ and PRESERVED_PLACEHOLDER_^^^^18sort_order18ti:\18^^^^, then

PRESERVED_PLACEHOLDER_^^^^18sort_order18 OR id:(Cho et al., 2023)18^^^^

(&&&^^^^18descending18^^^^&&&). 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 (&&&^^^^18max_results18sort_order18^^^^&&&). For three-dimensional unimodular and non-unimodular Lie groups with left-invariant Lorentzian metrics, the Ricci operator can have Segre types PRESERVED_PLACEHOLDER_^^^^18sort_order18max_results18^^^^, PRESERVED_PLACEHOLDER_^^^^18sort_order18sort_by18^^^^, and PRESERVED_PLACEHOLDER_^^^^18sort_order18submittedDate18^^^^, and the classification identifies flat metrics, constant-curvature metrics, semi-symmetric non locally symmetric metrics, and Ricci solitons (&&&^^^^18ti:\18max_results18^^^^&&&, &&&^^^^18ti:\18sort_by18^^^^&&&). Another algebraic example is furnished by cyclic Riemannian Lie groups, defined by

PRESERVED_PLACEHOLDER_^^^^18sort_order18sort_order18^^^^

where the cyclic condition constrains the Levi–Civita connection and curvature (&&&^^^^18max_results18ti:\18^^^^&&&).

On contact ^^^^18max_results18^^^^D Lie groups with a CR structure, the relevant invariants are Tanaka–Webster mean and Gauss curvature for embedded surfaces. If PRESERVED_PLACEHOLDER_^^^^18sort_order18descending18^^^^ denotes the Webster second fundamental form, then

PRESERVED_PLACEHOLDER_^^^^18sort_order18search_query18^^^^

(&&&^^^^18search_query18^^^^&&&). In the Heisenberg group, TW-cylindrical surfaces satisfy

PRESERVED_PLACEHOLDER_^^^^18sort_order18ti:\18^^^^

whereas tilted planes are TW-minimal with positive TW Gauss curvature away from the characteristic point (&&&^^^^18search_query18^^^^&&&). 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.

^^^^18sort_order18^^^^. 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 (&&&^^^^18 OR id:(Cho et al., 2023)18^^^^&&&, &&&^^^^18search_query18^^^^&&&, &&&^^^^18ti:\18^^^^&&&).

The second is integrability. Lie applicable surfaces carry a pencil of flat connections PRESERVED_PLACEHOLDER_^^^^18descending18search_query18^^^^, quadratic differential PRESERVED_PLACEHOLDER_^^^^18descending18ti:\18^^^^, Demoulin families, and Darboux transforms (&&&^^^^18ti:\18ti:\18^^^^&&&). Surfaces with spherical curvature lines are characterized by osculating sphere complexes PRESERVED_PLACEHOLDER_^^^^18descending18 OR id:(Cho et al., 2023)18^^^^ 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 (&&&^^^^18ti:\18 OR id:(Cho et al., 2023)18^^^^&&&). Constrained elastic curves themselves admit a Lie sphere characterization via polynomial conserved quantities of a family of flat connections PRESERVED_PLACEHOLDER_^^^^18descending18max_results18^^^^ (&&&^^^^18ti:\18 OR id:(Cho et al., 2023)18^^^^&&&).

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 (&&&^^^^18ti:\18search_query18^^^^&&&). In that framework, the sectional curvature of a source fibre is expressed by a groupoid version of the Arnold–Milnor “^^^^18ti:\18^^^^–^^^^18 OR id:(Cho et al., 2023)18^^^^–^^^^18max_results18^^^^– formula, and the Lie algebroid viewpoint subsumes Lie groups, principal bundles, and Riemannian submersion geometry within a single curvature formalism (&&&^^^^18ti:\18search_query18^^^^&&&).

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.