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Register a new userNon-naturally reductive Einstein metrics on $mathrm{SO}(n)$
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In this article, we prove that every compact simple Lie group $SO(n)$ for $ngeq 10$ admits at least $2left([frac{n-1}{3}]-2right)$ non-naturally reductive left-invariant Einstein metrics.
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Based on the representation theory and the study on the involutions of compact simple Lie groups, we show that $F_4$ admits non-naturally reductive Einstein metrics.
In this paper, the necessary and sufficient conditions for Matsumoto metrics $F=frac{alpha^2}{alpha-beta}$ to be Einstein are given. It is shown that if the length of $beta$ with respect to $alpha$ is constant, then the Matsumoto metric $F$ is an Einstein metric if and only if $alpha$ is Ricci-flat and $beta$ is parallel with respect to $alpha$. A nontrivial example of Ricci flat Matsumoto metrics is given.
We call a metric $m$-quasi-Einstein if $Ric_X^m$, which replaces a gradient of a smooth function $f$ by a vector field $X$ in $m$-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant metrics on simple Lie groups. First, we prove that $X$ is a left-invariant Killing vector field if the metric on a compact simple Lie group is $m$-quasi-Einstein. Then we show that every compact simple Lie group admits non-trivial $m$-quasi-Einstein metrics except $SU(3)$, $E_8$ and $G_2$, and most of them admit infinitely many metrics. Naturally, the study on $m$-quasi-Einstein metrics can be extended to pseudo-Riemannian case. And we prove that every compact simple Lie group admits non-trivial $m$-quasi-Einstein Lorentzian metrics and most of them admit infinitely many metrics. Finally, we prove that some non-compact simple Lie groups admit infinitely many non-trivial $m$-quasi-Einstein Lorentzian metrics.
We call a metric $m$-quasi-Einstein if $Ric_X^m$ (a modification of the $m$-Bakry-Emery Ricci tensor in terms of a suitable vector field $X$) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field $X$ such that the left-invariant Riemannian metric on a quadratic Lie group is $m$-quasi-Einstein is a Killing field. Then we construct infinitely many non-trivial $m$-quasi-Einstein metrics on solvable quadratic Lie groups $G(n)$ for $m$ finite.
In this paper, we first prove that homogeneous spaces $E_6/A_4$ and $E_6/A_1$ admit Einstein metrics which are $Ad(Ttimes A_1times A_4)$-invariant, and then show that they admit Non-Riemannian Einstein-Randers metrics.
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