Mathematicians solve 1965 geometry problem regarding Einstein metrics

Mathematicians Alberto Rodríguez Vázquez of the University of Santiago de Compostela and Gonzalo Cao Labora of the École Polytechnique Fédérale de Lausanne have resolved a geometric problem that has remained open since 1965.

The researchers successfully constructed new, non-homogeneous Einstein metrics on complex projective spaces. This discovery answers a question posed by French geometer Marcel Berger regarding whether other Einstein geometries existed beyond the known Fubini-Study metric. While Wolfgang Ziller identified a second geometry for certain odd dimensions in the 1980s, even dimensions had resisted solution for decades.

The new findings demonstrate the existence of these geometries for dimensions between 3 and 7, specifically providing the first known solutions for even dimensions n=4 and n=6. These Einstein metrics are significant due to their relationship with the equations of general relativity, which describe the geometry of space-time. Additionally, the complex projective space studied is a central object in modern geometry with potential applications in quantum computing.