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

Andreas Arvanitoyeorgos (Department of Mathematics, University of Patras, Rion, Greece)

Yusuke Sakane (Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan)

Marina Statha (Department of Mathematics, University of Patras, Rion, Greece)

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.

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