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Register a new userLocally conformally flat Lorentzian quasi-Einstein manifolds
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We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
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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 goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold $(M^{n},,g)$ with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstein fiber. In addition, the fiber has constant curvature if $n = 4.$
We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincare-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n geq 3$ and $(M^n , [h])$ is locally flat - namely $(M, h)$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits also a local situation and a global one. Furthermore the latter global situation includes the case of conformal infinities of Poincare-Einstein manifolds of dimension either 2 or of dimension greater than $2$ and which are locally flat, and hence the minimizing technique of Aubin- Schoen in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau which is not known to hold. Using the algebraic topological argument of Bahri-Coron, we bypass the latter positive mass issue and show that any conformal infinity of a Poincare-Einstein manifold of dimension either $n = 2$ or of dimension $n geq 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.
We find the index of $widetilde{ abla}$-quasi-conformally symmetric and $widetilde{ abla}$-concircularly symmetric semi-Riemannian manifolds, where $widetilde{ abla}$ is metric connection.
We classify quadruples $(M,g,m,tau)$ in which $(M,g)$ is a compact Kahler manifold of complex dimension $m>2$ with a nonconstant function $tau$ on $M$ such that the conformally related metric $g/tau^2$, defined wherever $tau e 0$, is Einstein. It turns out that $M$ then is the total space of a holomorphic $CP^1$ bundle over a compact Kahler-Einstein manifold $(N,h)$. The quadruples in question constitute four disjoint families: one, well-known, with Kahler metrics $g$ that are locally reducible; a second, discovered by Berard Bergery (1982), and having $tau e 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kahler surface metrics; and a fourth family, present only in odd complex dimensions $mge 9$. Our classification uses a {it moduli curve}, which is a subset $mathcal{C}$, depending on $m$, of an algebraic curve in $R^2$. A point $(u,v)$ in $mathcal{C}$ is naturally associated with any $(M,g,m,tau)$ having all of the above properties except for compactness of $M$, replaced by a weaker requirement of ``vertical compactness. One may in turn reconstruct $M,g$ and $tau$ from this $(u,v)$ coupled with some other data, among them a Kahler-Einstein base $(N,h)$ for the $CP^1$ bundle $M$. The points $(u,v)$ arising in this way from $(M,g,m,tau)$ with compact $M$ form a countably infinite subset of $mathcal{C}$.
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