The Journal of Geometric Analysis

, Volume 26, Issue 2, pp 996–1010

# Geometrically Formal Homogeneous Metrics of Positive Curvature

• Manuel Amann
• Wolfgang Ziller
Article

## Abstract

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.

## Keywords

Geometric formality Positive curvature Homogeneous spaces

## Mathematics Subject Classification

22F30 53C20 57T15

## Notes

### Acknowledgments

The first author was supported by IMPA and a research grant of the German Research Foundation DFG. The second author was supported by CAPES-Brazil, IMPA, the National Science Foundation and the Max Planck Institute in Bonn.

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