Notes on the Invariance of Tautness Under Lie Sphere Transformations

Abstract

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.