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# Geometry, Imaging and Computing

## Volume 2 (2015)

### Number 2

### New Einstein metrics on the Lie group $\mathrm{SO}(n)$ which are not naturally reductive

Pages: 77 – 108

DOI: http://dx.doi.org/10.4310/GIC.2015.v2.n2.a1

#### Authors

#### Abstract

We obtain new invariant Einstein metrics on the compact Lie groups $\mathrm{SO}(n) \; (n \geq 7)$ which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on $\mathrm{SO}(n)$ and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.