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Thomas E. Cecil

Publications and source records attributed to Thomas E. Cecil.

11 recordsLinked to original sources
arXiv2026

Taut and Dupin Submanifolds (Updated Version)

This is an updated version of a paper by the author which originally appeared in 1997. The original paper was a survey of the closely related fields of taut and Dupin submanifolds of Euclidean space, and this updated version includes many results in the field that have appeared since the publication of the original version. The emphasis is on stating results in their proper context and noting areas for future research, and relatively few proofs are given. The important class of isoparametric hypersurfaces is surveyed in detail, as is the relationship between the two concepts of taut and Dupin. Also included is a brief introduction to submanifold theory in Lie sphere geometry which is needed to state many known results on Dupin submanifolds accurately. The paper concludes with detailed descriptions of the main known classification results for both Dupin and taut submanifolds.

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arXiv2025

Tight Spherical Embeddings (Updated Version)

This is an updated version of a paper which appeared in the proceedings of the 1979 Berlin Colloquium on Global Differential Geometry. This paper contains the original exposition together with some notes by the authors made in 2025 (as indicated in the text) that give references to descriptions of progress made in the field since the time of the original version of the paper. The main result of this paper is that every compact isoparametric hypersurface $M^n \subset S^{n+1} \subset {\bf R}^{n+2}$ is tight, i.e., every non-degenerate linear height function $\ell_p$, $p \in S^{n+1}$, has the minimum number of critical points on $M^n$ required by the Morse inequalities. Since $M^n$ lies in the sphere $S^{n+1}$, this implies that $M^n$ is also taut in $S^{n+1}$, i.e., every non-degenerate spherical distance function has the minimum number of critical points on $M^n$. A second result is that the focal submanifolds of isoparametric hypersurfaces in $S^{n+1}$ must also be taut. The proofs of these results are based on Münzner's fundamental work on the structure of a family of isoparametric hypersurfaces in a sphere.

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arXiv2025

Constructions of Compact Dupin Hypersurfaces with Non-constant Lie Curvatures

A hypersurface $M$ in the unit sphere $S^n \subset {\bf R}^{n+1}$ is Dupin if along each curvature surface of $M$, the corresponding principal curvature is constant. If the number $g$ of distinct principal curvatures is constant on $M$, then $M$ is called proper Dupin. In this expository paper, we give a detailed description of two important types of constructions of compact proper Dupin hypersurfaces in $S^n$. One construction was published in 1989 by Pinkall and Thorbergsson, and the second was published in 1989 by Miyaoka and Ozawa. Both types of examples have the property that they do not have constant Lie curvatures (Lie invariants discovered by Miyaoka), which are the cross-ratios of the principal curvatures, taken four at a time. Thus, these examples are not equivalent by a Lie sphere transformation to an isoparametric (constant principal curvatures) hypersurface in $S^n$. So they are counterexamples to a conjecture of Cecil and Ryan in 1985 that every compact proper Dupin hypersurface in $S^n$ is equivalent to an isoparametric hypersurface by a Lie sphere transformation.

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arXiv2025

Notes on the Invariance of Tautness Under Lie Sphere Transformations

An embedding $ϕ:V \rightarrow S^n$ of a compact, connected manifold $V$ into the unit sphere $S^n \subset {\bf R}^{n+1}$ is said to be taut, if every nondegenerate spherical distance function $d_p$, $p \in S^n$, is a perfect Morse function on $V$, i.e., it has the minimum number of critical points on $V$ required by the Morse inequalities. In these notes, we give an exposition of the proof of the invariance of tautness under Lie sphere transformations due to Álvarez Paiva. First we extend the definition of tautness of submanifolds of $S^n$ to the concept of Lie-tautness of Legendre submanifolds of the contact manifold $Λ^{2n-1}$ of projective lines on the Lie quadric $Q^{n+1}$. This definition has the property that if $ϕ:V \rightarrow S^n$ is an embedding of a compact, connected manifold $V$, then $ϕ(V)$ is a taut submanifold in $S^n$ if and only if the Legendre lift $λ$ of $ϕ$ is Lie-taut. Furthermore, Lie-tautness is invariant under the action of Lie sphere transformations on Legendre submanifolds. As a consequence, we get that if $ϕ:V \rightarrow S^n$ and $ψ:V \rightarrow S^n$ are two embeddings of a compact, connected manifold $V$ into $S^n$, such that their corresponding Legendre lifts are related by a Lie sphere transformation, then $ϕ$ is a taut embedding if and only if $ψ$ is a taut embedding. Thus, in that sense, tautness is invariant under Lie sphere transformations. The key idea is to formulate tautness in terms of real-valued functions on $S^n$ whose level sets form a parabolic pencil of unoriented spheres in $S^n$, and then show that this is equivalent to the usual formulation of tautness in terms of spherical distance functions, whose level sets in $S^n$ form a pencil of unoriented concentric spheres.

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arXiv2024

On the Work of Cartan and Münzner on Isoparametric Hypersurfaces

A hypersurface $M^n$ in a real space form ${\bf R}^{n+1}$, $S^{n+1}$, or $H^{n+1}$ is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and Münzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938--1940, and two papers of Münzner that were published in preprint form in the early 1970's, and as journal articles in 1980--1981. These papers of Cartan and Münzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres.

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arXiv2022

Classifications of Dupin Hypersurfaces in Lie Sphere Geometry

This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of $S^n$ (or ${\bf R}^n$), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

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arXiv2021

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in ${\bf R}^n$

A hypersurface $M$ in ${\bf R}^n$ or $S^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on $M$, i.e., the number of distinct principal curvatures is constant on $M$. The notions of Dupin and proper Dupin hypersurfaces in ${\bf R}^n$ or $S^n$ can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in ${\bf R}^n$ or $S^n$, including proofs of several fundamental results.

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arXiv2021

Compact Dupin Hypersurfaces

A hypersurface $M$ in ${\bf R}^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on $M$, i.e., each continuous principal curvature function has constant multiplicity on $M$. These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in ${\bf R}^n$ or $S^n$. The theory of compact proper Dupin hypersurfaces in $S^n$ is closely related to the theory of isoparametric hypersurfaces in $S^n$, and many important results in this field concern relations between these two classes of hypersurfaces. In 1985, Cecil and Ryan conjectured on p. 184 of the book, "Tight and Taut Immersions of Manifolds," that every compact, connected proper Dupin hypersurface $M \subset S^n$ is equivalent to an isoparametric hypersurface in $S^n$ by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.

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arXiv2020

Dupin Submanifolds in Lie Sphere Geometry (updated version)

A hypersurface $M^{n-1}$ in Euclidean space $E^n$ is proper Dupin if the number of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together with some remarks by T.Cecil made in 2020 (as indicated in the text) that describe progress in the field since the time of the original version, as well as some important remaining open problems in the field.

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arXiv2008

Isoparametric and Dupin Hypersurfaces

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Elie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere $S^n$. A hypersurface $M^{n-1}$ in a real space-form is proper Dupin if the number $g$ of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.

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