A characterization of the n-dimensional torus via intrinsically harmonic forms

TL;DR

This paper characterizes the n-dimensional torus as the unique closed manifold supporting a specific set of closed 1-forms whose product defines a non-zero cohomology class, refining previous characterizations.

Contribution

It improves existing results by showing the torus's uniqueness with a set of (n-1) closed 1-forms and a non-zero cohomological product.

Findings

The n-torus is uniquely characterized by supporting (n-1) closed 1-forms with a non-zero cohomology class.

The result refines the understanding of the torus's differential form structure.

Provides a new criterion for identifying the n-torus among closed manifolds.

Abstract

The $n$-torus is the the unique closed manifold supporting a set of $n$ linearly independent closed $1$-forms. In this paper we improve on this result and show that the torus is the unique closed $n$-dimensional manifold supporting a linearly independent set consisting of $(n-1)$ closed $1$-forms whose product determines a non-zero cohomological class.