Local classification of proper Dupin hypersurfaces in $E^n$ for $n \geq 5$ up to Lie sphere equivalence

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A hypersurface of Euclidean space is Dupin if each principal curvature is constant along the leaves of its principal foliation, and proper Dupin if, in addition, the number $g$ of distinct principal curvatures is constant. Cecil and Chern (1989) developed a moving-Lie-frame method in Lie sphere geometry and used it to re-derive Pinkall's local classification of Dupin hypersurfaces in $E^4$, explicitly stating that no comparable classification was known for $E^n$, $n \geq 5$, and expressing the hope that their method would extend. The irreducible case with three principal curvatures was classified in all dimensions by Cecil and Jensen (1998), and the case of four principal curvatures is classified only under strong multiplicity and constant-Lie-curvature restrictions (Cecil and Jensen 2000; Cecil, Chi and Jensen 2007). Still open are: the classification of reducible Dupin hypersurfaces with three principal curvatures in $n \geq 5$, the four-curvature case with general multiplicities or nonconstant Lie curvature, the six-curvature case, the compact classification conjectures for $g = 4$ and $g = 6$, and any framework covering loci where $g$ is not constant. The surviving open core is a local classification, up to Lie sphere equivalence, of proper Dupin hypersurfaces in $E^n$ ($n \geq 5$) comparable in completeness to the known $E^4$ classification.

Background

For a hypersurface $M^{n-1}$ in Euclidean space $E^n$, the principal curvatures are the eigenvalues of the shape operator (the derivative of the unit normal field along tangent directions). A connected submanifold of $M$ whose tangent spaces are principal spaces at every point is called a curvature surface. The hypersurface is Dupin if each principal curvature is constant along the leaves of its corresponding principal foliation (the foliation of $M$ by the curvature surfaces on which that principal curvature is defined), and proper Dupin if in addition the number $g$ of distinct principal curvatures is constant on $M$. The classical examples in $E^3$ are the cyclides of Dupin, surfaces whose two principal curvatures are both constant along their curvature lines.

The theory is naturally embedded in Lie sphere geometry, introduced by Sophus Lie. Oriented hyperspheres, oriented hyperplanes, and points of $E^n \cup \{\infty\} = S^n$ correspond bijectively to the points of a quadric $Q^{n+1}$ in projective space $P^{n+2}$ defined by an inner product of signature $(n+1,2)$ on $\mathbb{R}^{n+3}$. The Lie sphere group $O(n+1,2)/\pm I$ is the group of contact transformations of the projectivized cotangent bundle that carries oriented hyperspheres to oriented hyperspheres; it contains the Moebius (conformal) group and the Euclidean group. An oriented hypersurface induces a Legendre submanifold $\lambda: B^{n-1} \to \Lambda^{2n-1}$ into the manifold $\Lambda^{2n-1}$ of projective lines lying on $Q^{n+1}$ (its Legendre lift), one line for each contact element. The focal points of $\lambda$ along each line are its curvature spheres; for a hypersurface of $E^n$ or $S^n$ they correspond exactly to the distinct principal curvatures. The Dupin and proper Dupin conditions are preserved by stereographic projection, so the theory is essentially the same for hypersurfaces of $E^n$ and of the sphere $S^n$, and both are studied through their Legendre lifts. The Dupin condition — each curvature sphere constant along its curvature surfaces — is invariant under the Lie sphere group, and two Dupin hypersurfaces are Lie equivalent when their Legendre lifts are related by a Lie sphere transformation. For $g = 4$ and $g = 6$, the basic Lie invariants are the Lie curvatures, the cross-ratios of the curvature spheres (equivalently, of the principal curvatures) taken four at a time, introduced by Miyaoka; for $g = 3$ in $E^4$ the corresponding scalar invariant is denoted $\rho$ in Cecil and Chern's treatment.

Two facts shape the classification problem. First, by a theorem of Singley, $g$ is locally constant on a dense open subset of any Legendre submanifold, but it need not be constant everywhere; Cecil and Chern therefore work exclusively on open sets where $g$ is constant, and they exhibit the Legendre lift of a tube over a torus $T^2 \subset E^3 \subset E^4$ as an example where $g$ drops. Second, locally (non-compactly) there is no restriction on $g$: Pinkall's standard constructions — taking cylinders, hypersurfaces of revolution, and tubes of constant radius over a Dupin hypersurface of lower dimension (a fourth, cone construction is Lie equivalent to the tube one) — produce proper Dupin hypersurfaces in $E^n$ with any number $g$ of distinct principal curvatures and any prescribed multiplicities. A Dupin hypersurface is reducible if it is locally Lie equivalent to the result of one of these constructions, and irreducible otherwise. In the compact case the situation is rigid: Thorbergsson proved that a compact, connected, properly embedded Dupin hypersurface in $S^n$ is taut (every nondegenerate distance function has the minimum number of critical points allowed by the Morse inequalities), which forces $g \in \{1,2,3,4,6\}$, the same restriction as for isoparametric hypersurfaces — hypersurfaces of $S^n$ all of whose principal curvatures are constant.

The model results are the classifications in low dimensions. Proper Dupin hypersurfaces with $g = 2$ are the cyclides, classified up to Lie equivalence by Pinkall. For $E^4$, Pinkall's dissertation (published 1985) obtained a local classification up to Lie equivalence of Dupin hypersurfaces with three principal curvatures: if the invariant $\rho$ is nonzero, every connected such hypersurface is contained in a unique compact one, and any two are locally Lie equivalent to an open subset of Cartan's isoparametric hypersurface $M^3 \subset S^4$ — the unique compact model, a tube of constant radius over a Veronese surface (the standard projective embedding of $\mathbb{R}P^2$) in $S^4$; if $\rho = 0$, the hypersurface is reducible, obtained from a cyclide in $E^3$ by the standard constructions, and Pinkall also classified these up to Lie equivalence. Cecil and Chern (1989) re-derived this classification by the method of moving Lie frames — normalizing a Lie frame along the Legendre lift, reducing the Maurer–Cartan structure equations (the compatibility conditions encoding how the moving frame changes along the submanifold), and integrating the resulting completely integrable differential systems — and stated explicitly that this was the frontier: no comparable classification existed for $E^n$, $n \geq 5$, and the moving-frame framework was offered as the direction for extending it.

Problem Statement

Obtain a local classification, up to Lie sphere equivalence, of proper Dupin hypersurfaces in $E^n$ for $n \geq 5$, comparable in completeness and explicitness to the classification of Dupin hypersurfaces in $E^4$ obtained by Pinkall and re-derived with moving Lie frames by Cecil and Chern. Specifically: for every connected proper Dupin hypersurface in $E^n$ (or equivalently $S^n$) with $n \geq 5$ — irreducible or reducible, with any number $g$ of distinct principal curvatures and any multiplicities — determine explicit local normal forms or model hypersurfaces, and prove that every such hypersurface is locally Lie equivalent to exactly one model (completeness and non-redundancy). Within this objective, the partial results and remaining open cases — the irreducible case with $g = 3$, known by Cecil and Jensen (1998), and the still-open reducible case with $g = 3$ in $n \geq 5$, the case $g = 4$ with general multiplicities or nonconstant Lie curvature, and the case $g = 6$ — all concern this local classification. The compact classification conjecture for $g = 4$ and $g = 6$ with constant Lie curvatures, and the study of loci on which the number $g$ of distinct curvature spheres is not constant, are related open problems, but they lie outside the proper-Dupin local classification requested here.

References

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