In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M, g) is compact. Established in the locally conformally flat case by Schoen [43, 44] and for n ≤ 24 by Khuri– Marques–Schoen [26], it has revealed to be generally false for n ≥ 25 as shown by Brendle [8] and Brendle–Marques [9]. A stronger version of it...

In this paper, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton. 1. The result A complete Riemannian metric gij on a smooth manifold M n is called a gradient steady Ricci soliton if there exists a smooth function F on M such that the Ricci tensor Rij of the metric gij is given by the Hessian of F : Rij = ∇i∇jF. (1....

We show that complete conformally flat manifolds of dimension n > 3 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to R n or a spherical spaceform Sn/Γ. This extends previous results due to Q.-M. Cheng and B.-L. Chen and X.-P. Zhu. In this note, we study compl...

It is given a complete study of locally conformally flat metrics satisfying some weakly-Einstein conditions. shown that they are either product Mn(c)×Mn(−c) or warped R×fRn−1 for specific warping function. Moreover, conditions on fixed points the RG2 flow pointed out.

In this paper we are interested in the multiplicity of the Yamabe metrics for compact Riemannain manifolds. This problem may be approached only when the scalar curvature is positive. This is the case notably of certain locally conformally flat manifolds. Furthermore, there exists a link with the singular Yamabe problem. We give various applications of this study.

Prescribing, by conformal transformation, the k-elementary symmetric polynomial of the Schouten tensor σk(P) to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for 3 ≤ k ≤ n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.

In this paper, we study the Cartan space with some (α, β) metrics, in particular Randers metric admitting h-metrical d-connection. Further, we show that the condition for Cartan space with Randers metric to be locally Minkowski and Conformally flat. 2000 Mathematics Subject Classification: 53B40, 53C60