Transverse geometric formality

TL;DR

This paper introduces the concept of transverse geometric formality in Riemannian foliations, exploring its implications for the geometry and topology of the foliated manifold.

Contribution

It defines transversely geometrically formal metrics and investigates their geometric and topological consequences in Riemannian foliations.

Findings

Transversely geometrically formal metrics impose strong topological restrictions.

Such metrics influence the harmonic form structure on foliated manifolds.

The paper establishes new links between foliation geometry and harmonic analysis.

Abstract

A Riemannian metric on a closed manifold is said to be geometrically formal if the wedge product of any two harmonic forms is harmonic; equivalently, the interior product of any two harmonic forms is harmonic. Given a Riemannian foliation on a closed manifold, we say that a bundle-like metric is transversely geometrically formal if the interior product of any two basic harmonic forms is basic harmonic. In this paper, we examine the geometric and topological consequences of this condition.