[Paper Review] A surprising fibration of S3 x S3 by great 3-spheres

Haggai Nuchi|arXiv (Cornell University)|Jul 17, 2014

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper presents a new fiberwise homogeneous fibration of the Clifford torus $S^3 \times S^3$ in $S^7$ by great 3-spheres that are not parallel to one another, distinguishing it from the classical Hopf fibration. Using a non-constant distance-decreasing map $f$ in Petro's moduli space, the authors construct fibrations where isometries preserve fiber structure but fibers rotate relative to each other, revealing a novel geometric configuration with linked hot and cold circles on each fiber.

ABSTRACT

In this paper, we describe a new surprising example of a fibration of the Clifford torus S3 x S3 in the 7-sphere by great 3-spheres, which is fiberwise homogeneous but whose fibers are not parallel to one another. In particular it is not part of a Hopf fibration. A fibration is fiberwise homogeneous when for any two fibers there is an isometry of the total space taking fibers to fibers and taking the first given fiber to the second one. We also describe in detail the geometry of this surprising fibration and how it differs from the Hopf fibration.

Motivation & Objective

• To construct and analyze a new class of fibrations of $S^3 \times S^3$ by great 3-spheres that are fiberwise homogeneous but not parallel, distinguishing them from the Hopf fibration.
• To demonstrate that such fibrations arise from non-constant distance-decreasing maps $f: S^3 \to S^3$ in John Petro's moduli space of fibrations.
• To characterize the geometric behavior of fibers, particularly the evolution of hot and cold circles on a fixed fiber as neighboring fibers rotate around it.
• To show that the symmetry group acts transitively on fibers, confirming fiberwise homogeneity despite non-parallelism.
• To clarify how this fibration differs fundamentally from the Hopf fibration in terms of fiber alignment, symmetry, and geometric structure.

Proposed method

• The fibration is constructed using graphs of isometries $x \mapsto p x f(p)^{-1}$, where $f$ is a distance-decreasing map from $S^3$ to $S^3$, parameterized by $p \in S^3$.
• The authors use the identification of $S^3$ with unit quaternions and $SO(4) \cong S^3 \times S^3 / \mathbb{Z}_2$ to model isometries as left and right multiplication.
• They apply a sequence of isometries $T$ and $T'$ to align a moving fiber $\Sigma_{p(\theta)}$ to a standard fiber $\Delta$, enabling analysis of relative position via first-order approximations as $\varepsilon \to 0$.
• By analyzing the limit of $q'' \sim 1 + 2\varepsilon \sin\alpha \cdot j_\theta$, they determine the direction of closest and furthest points between fibers.
• They identify the 'hot' and 'cold' circles on a fixed fiber $\Sigma_1$ as the sets of points closest to and furthest from a rotating fiber $\Sigma_{p(\theta)}$, respectively.
• The hot and cold circles are shown to spin around fixed antipodal points on $\Sigma_1$ as $\theta$ varies, forming linked 2-spheres.

Experimental results

Research questions

• RQ1 Can a fibration of $S^3 \times S^3$ by great 3-spheres be fiberwise homogeneous without having parallel fibers?
• RQ2 How does the geometry of a non-parallel, fiberwise homogeneous fibration differ from that of the Hopf fibration?
• RQ3 What is the behavior of the closest and furthest points between two non-parallel great 3-sphere fibers in $S^3 \times S^3$?
• RQ4 How do the hot and cold circles on a fixed fiber evolve as neighboring fibers rotate around it?
• RQ5 What role does the non-constant map $f$ play in generating a fibration that is not a restriction of the Hopf fibration?

Key findings

• The fibration is fiberwise homogeneous: for any two fibers, there exists an isometry of $S^3 \times S^3$ mapping one fiber to the other while preserving the fibration structure.
• The fibers are not parallel; they rotate relative to one another as $\theta$ varies, with the closest and furthest points forming rotating great circles on each fiber.
• The set of points on $\Sigma_1$ closest to $\Sigma_{p(\theta)}$ forms a great circle passing through $e^{i(\alpha/2 - \pi/4)}$ and $(j\cos\theta + k\sin\theta)e^{i(\alpha/2 - \pi/4)}$, independent of $\theta$.
• The set of points furthest from $\Sigma_{p(\theta)}$ lies on a great circle through $i e^{i(\alpha/2 - \pi/4)}$ and $(-j\sin\theta + k\cos\theta)e^{i(\alpha/\!2 - \pi/4)}$, also independent of $\theta$.
• The hot and cold circles spin around fixed antipodal points on $\Sigma_1$ as $\theta$ varies, forming linked 2-spheres reminiscent of eggbeater blades.
• The fibration is not a restriction of the Hopf fibration, as the Hopf fibration corresponds only to the constant map $f \equiv 1$, while this construction uses a non-constant $f$.