# On geometric formality of rationally elliitic manifolds in dimensions $6$ and $7$

## TL;DR

This paper investigates the geometric formality of rationally elliptic manifolds in dimensions 6 and 7, establishing conditions under which these manifolds have cohomology similar to symmetric spaces.

## Contribution

It proves that certain six- and seven-dimensional rationally elliptic manifolds are cohomologically equivalent to symmetric spaces, clarifying their geometric structure.

## Key findings

- Six-dimensional biquotients with b2=3 have cohomology of symmetric spaces.
- Rationally hyperbolic six-dimensional manifolds with b2≤2 and b3=0 are not geometrically formal.
- Seven-dimensional geometrically formal rationally elliptic manifolds also have cohomology of symmetric spaces.

## Abstract

We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and $b_3=0$ can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic manifolds have the real cohomology of symmetric spaces as well.