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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.
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This paper contributes to the study of the Matsumoto metric F=alpha^2/beta, where the alpha is a Riemannian metric and the beta is a one form. It is shown that such a Matsumoto metric F is of scalar flag curvature if and only if F is projectively flat.
We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension $geq 8$.
In this paper, we classify three-locally-symmetric spaces for a connected, compact and simple Lie group. Furthermore, we give the classification of invariant Einstein metrics on these spaces.
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.
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.
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