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

Abstract

We obtain new invariant Einstein metrics on the compact Lie groups SO(n) (n ≥ 7) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on 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.

Non-naturally reductive Einstein metrics on $$\mathrm {SO}(n)$$ SO ( n )

“…In [2], the authors prove that, for any n ≥ 9, the Lie group SO(n) admits at least one left- We will prove Theorem 3.1 by solving homogeneous Einstein equations r 1 = r 2 , r 2 = r 3 , r 3 = r 12 , r 12 = r 13 , r 13 = r 23 under the assumption k = k 1 = k 2 and l = k 3 . Furthermore, we consider the metric (2.4) with x 1 = x 2 .…”

“…Furthermore, they prove that SO(n) admits non-naturally reductive Einstein metrics which are Ad(SO(n − 6) × SO(3) × SO(3))-invariant. In section 3, based on the Ricci tensor formulae in [2] and the technique of Gröbner basis, we prove that SO(2k + l) admits at least two non-naturally reductive Einstein metrics which are Ad(SO(k) × SO(k) × SO(l))-invariant when l > k ≥ 3. It implies Theorem 1.1.…”

“…After that, Chen and Liang [7] give one non-naturally reductive left-invariant Einstein metric on F 4 . In [2] Yan and Deng prove an interesting result in [17]: for any integer n = p l 1 1 p l 2 2 · · · p ls s with p i prime and p i = p j , (1) SO(2n) admits at least (l 1 +1)(l 2 +1) · · · (l s +1)−3 non-equivalent non-naturally reductive Einstein metrics, (2) and Sp(2n) admits at least (l 1 + 1)(l 2 + 1) · · · (l s + 1) − 1 non-equivalent non-naturally reductive Einstein metrics.…”

“…The paper is organized as follows. In section 2, we recall the study on non-naturally reductive left-invariant metrics on SO(n) in [2], in particular the Ricci tensor of a class of left-invariant metrics ·, · on SO(n), and sufficient and necessary conditions for ·, · to be naturally reductive.…”

“…[2]). If a left-invariant metric ·, · of the form (2.4) on G is naturally reductive with respect to G × L, where L is a closed subgroup of G = SO(k 1 + k 2 + k 3 ), then one of the following holds:…”