Notes on the Invariance of Tautness Under Lie Sphere Transformations

Authors

Thomas E. Cecil, College of the Holy CrossFollow

Document Type

Article

Publication Date

6-10-2025

Department

Mathematics

Abstract

An embedding ϕ : V → Sⁿ of a compact, connected manifold V into the unit sphere Sⁿ ⊂ Rⁿ⁺¹ is said to be taut, if every nondegenerate spherical distance function d_p, p ∈ Sⁿ, 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 ´Alvarez Paiva. First we extend the definition of tautness of submanifolds of Sⁿ to the concept of Lie-tautness of Legendre submanifolds of the contact manifold Λ²ⁿ⁻¹ of projective lines on the Lie quadric Qⁿ⁺¹. This definition has the property that if ϕ : V → Sⁿ is an embedding of a compact, connected manifold V , then ϕ(V ) is a taut submanifold in Sⁿ if and only if the Legendre lift λ of ϕ is Lietaut. Furthermore, Lie-tautness is invariant under the action of Lie sphere transformations on Legendre submanifolds. As a consequence, we get that if ϕ : V → Sⁿ and ψ : V → Sⁿ are two embeddings of a compact, connected manifold V into Sⁿ, 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 Sⁿ whose level sets form a parabolic pencil of unoriented spheres in Sⁿ, and then show that this is equivalent to the usual formulation of tautness in terms of spherical distance functions, whose level sets in Sⁿ form a pencil of unoriented concentric spheres.

Comments

Mathematics Subject Classification numbers are: 53A07, 53A40, 53B25, 53C40, 53C42

DOI

https://doi.org/10.48550/arXiv.2506.08834

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Repository Citation

Cecil, Thomas E., "Notes on the Invariance of Tautness Under Lie Sphere Transformations" (2025). Mathematics and Computer Science Department Faculty Scholarship. 23.
https://crossworks.holycross.edu/math_fac_scholarship/23