Non-trivial $m$-quasi-Einstein metrics on quadratic Lie groups

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.

Non-naturally reductive Einstein metrics on the compact simple Lie group $F_4$

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.

Non-trivial $m$-quasi-Einstein metrics on simple Lie groups

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.

The CIV Baldwin effect in QSOs from Seventh Data Release of the Sloan Digital Sky Survey

Using the properties of SDSS DR7 QSOs catalog from Shen et al., the Baldwin effect, its slope evolution, the underlying drive for a large sample of 35019 QSOs with reliable spectral analysis are investigated. We find that the Baldwin effect exists in this large QSOs sample, which is almost the same in 11 different redshift bins, up to $zsim 5$. The slope is -0.238 by the BCES (civ EW depends on the continuum), -0.787 by the BCES bisector. For 11 redshift-bins, there is an increasing of the Baldwin effect slope from $zsim1.5$ to $zsim2.0$. From $zsim2.0$ to $zsim5.0$, the slope change is not clear considering their uncertainties or larger redshift bins. There is a strong correlation between the rest-frame civ EW and civ-based mbh while the relation between the civ EW and mgii-based mbh is very weak. With the correction of civ-based mbh from the civ blueshift relative to mgii, we suggest that this strong correlation is due to the bias of the civ-based mbh, with respect to that from the mgii line. Considering the mgii-based mbh, a medium strong correlation is found between the civ EW and the Eddington ratio, which implies that the Eddington ratio seems to be a better underlying physical parameter than the central black hole mass.