Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRicci-flat and Einstein pseudoriemannian nilmanifolds
71
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This is partly an expository paper, where the authors work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $leq8$ are obtained. Some related open questions are presented.
rate research
Read More
We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $leq6$, every nice nilpotent Lie group of dimension $leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${rm SL}(n)$, ${rm SO}(p,q)$, ${rm Sp}(n,mathbb R)$. Most of these metrics are shown not to be flat.
It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson-Walker warped product, if the gradient of the potential function is non null, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.
The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless $kappa$-Einstein soliton and is realized on the cotangent bundle of an affine surface.
In this work, we consider a class of Finsler metrics using the warped product notion introduced by Chen, S. and Zhao (2018), with another warping, one that is consistent with static spacetimes. We will give the PDE characterization for the proposed metrics to be Ricci-flat and explicitly construct two non-Riemannian examples.
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of $X$ over its canonical model $X_{can}$. This assumption on the Ricci curvature always holds when the Kodaira dimension of $X$ is $n$, $n-1$ or when the general fibre of $X$ over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to its canonical model when $X$ has Kodaira dimension $1$ with $K_X$ being semi-ample and the general fibre of $X$ over its canonical model being a complex torus. We also prove the Gromov-Hausdorff limit of collapsing Ricci-flat Kahler metrics on a holomorphically fibred Calabi-Yau manifold is unique and is homeomorphic to the metric completion of the corresponding twisted Kahler-Einstein metric on the regular part of its base.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
