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Basic Albanese maps of regular Riemannian foliations

Kinga Słowik, Robert Wolak

TL;DR

The paper defines the basic Albanese map for foliated Riemannian manifolds via basic 1-forms and connects it to the classical Albanese map through the first basic cohomology $H^1(M/\mathcal{F})$. It develops a cohomological framework using Molino theory to relate rational and integer basic first cohomology, proves universal and harmonic properties of the basic Albanese map, and analyzes the submersion case and nilmanifold applications. It also extends to singular Riemannian foliations, showing that the dimension of $H^1(M/\mathcal{F})$ controls the stratified codimensions of leaf closures and constrains possible leaf behaviors. Together these results provide a robust tool for understanding foliated geometry via basic cohomology and harmonic maps to flat tori, with concrete examples on nilmanifolds and Iwasawa-type manifolds.

Abstract

In the paper we introduce the notion of basic Albanese map which we define for foliated Riemannian manifolds using basic 1-forms. We relate this mapping to the classical Albanese map for the ambient manifold. The study of general properties is supplemented with the description of several important examples.

Basic Albanese maps of regular Riemannian foliations

TL;DR

The paper defines the basic Albanese map for foliated Riemannian manifolds via basic 1-forms and connects it to the classical Albanese map through the first basic cohomology . It develops a cohomological framework using Molino theory to relate rational and integer basic first cohomology, proves universal and harmonic properties of the basic Albanese map, and analyzes the submersion case and nilmanifold applications. It also extends to singular Riemannian foliations, showing that the dimension of controls the stratified codimensions of leaf closures and constrains possible leaf behaviors. Together these results provide a robust tool for understanding foliated geometry via basic cohomology and harmonic maps to flat tori, with concrete examples on nilmanifolds and Iwasawa-type manifolds.

Abstract

In the paper we introduce the notion of basic Albanese map which we define for foliated Riemannian manifolds using basic 1-forms. We relate this mapping to the classical Albanese map for the ambient manifold. The study of general properties is supplemented with the description of several important examples.
Paper Structure (11 sections, 16 theorems, 38 equations)

This paper contains 11 sections, 16 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integer and rational basic cohomology for Riemannian foliations
  4. The case of transversely parallelizable foliations
  5. General case
  6. Dense leaves
  7. Basic Albanese map
  8. Construction and basic properties
  9. Submersion case
  10. Application to foliations on nilmanifolds
  11. Some applications to singular Riemannian foliations

Key Result

Lemma 2.1

Let $X$ be a vector field on $M$. Then the following conditions are equivalent:

Theorems & Definitions (41)

  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4: Molino structure theorem, MolinoTond
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • ...and 31 more