Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Riemannian foliations on CROSSes

Marco Radeschi, Lorenzo Scoffone

TL;DR

The paper addresses the problem of classifying Riemannian foliations on manifolds homeomorphic to CROSSes, building on sphere results to CP^n, HP^n, and OP^2. It uses Hopf-like pullbacks to reduce to leaf-dimension constraints, then applies Browder–Escobales, Gromoll–Grove, Novikov, and Stiefel–Whitney tools to rule out most cases and identify a canonical foliation in the CP^{2m+1} canonical-metric setting as the twistor fibration T: CP^{2m+1} → HP^m; it also shows there are no nontrivial foliations on HP^n or OP^2. The main contributions are: (i) foliations on CP^{2m+1} arise from S^2-bundles with base cohomology like HP^m; (ii) for the canonical metric, the foliation is congruent to the twistor fibration; (iii) nonexistence results for HP^n and OP^2; and (iv) partial results for spaces homotopy equivalent to CROSSes. These results provide strong geometric and topological constraints on foliations in CROSS-like manifolds and identify a canonical geometric structure (twistor fibration) governing such foliations on CP^{2m+1}.

Abstract

We classify Riemannian foliations of manifolds homeomorphic to CROSSes.

Riemannian foliations on CROSSes

TL;DR

The paper addresses the problem of classifying Riemannian foliations on manifolds homeomorphic to CROSSes, building on sphere results to CP^n, HP^n, and OP^2. It uses Hopf-like pullbacks to reduce to leaf-dimension constraints, then applies Browder–Escobales, Gromoll–Grove, Novikov, and Stiefel–Whitney tools to rule out most cases and identify a canonical foliation in the CP^{2m+1} canonical-metric setting as the twistor fibration T: CP^{2m+1} → HP^m; it also shows there are no nontrivial foliations on HP^n or OP^2. The main contributions are: (i) foliations on CP^{2m+1} arise from S^2-bundles with base cohomology like HP^m; (ii) for the canonical metric, the foliation is congruent to the twistor fibration; (iii) nonexistence results for HP^n and OP^2; and (iv) partial results for spaces homotopy equivalent to CROSSes. These results provide strong geometric and topological constraints on foliations in CROSS-like manifolds and identify a canonical geometric structure (twistor fibration) governing such foliations on CP^{2m+1}.

Abstract

We classify Riemannian foliations of manifolds homeomorphic to CROSSes.
Paper Structure (8 sections, 3 theorems, 11 equations)

This paper contains 8 sections, 3 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. $M$ homeomorphic to $\mathbb{CP}^{n}$
  3. Case 1: $\bar{\mathcal{F}}$ has $3$-dimensional fibres.
  4. Case 2: $\bar{\mathcal{F}}$ has $7$-dimensional fibres.
  5. $\mathbb{CP}^{2n+1}$ with its canonical metric
  6. $M$ homeomorphic to $\mathbb{HP}^{n}$
  7. $M$ homeomorphic to $\mathbb{O}\mathbb{P}^{2}$
  8. Notes on manifolds homotopy equivalent to CROSSes

Key Result

Theorem 1.1

Consider a $k$-dimensional Riemannian foliation $\mathcal{F}$ of a Riemannian manifold $(M,g)$ homeomorphic to $\mathbb{S}^{n}$. Assuming $0<k<n$, one of the following holds: Furthermore all these cases can occur.

Theorems & Definitions (4)

  • Theorem 1.1: lytchakwilking
  • Theorem A
  • Lemma 3.1
  • proof