We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that M#N admits a conformally flat Riemannian metric.

We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is nonnegative when the manifold is locally conformally flat and the σk curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, near infinity, the manifold is isometric to a Euclidean end.

The study of weakly symmetric and weakly projective symmetric manifold were initiated by Tamassy and Binh in 1989. Later on several authors studied weakly symmetric Riemannian manifold and analogous structures, viz. Weakly Ricci symmetric, weakly projective symmetric and weakly conformally symmetric Riemannian manifolds. At first we cited examples of weakly symmetric Riemannian manifolds. Next ...

A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 tha...

This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from Rn by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold S × S, conformally flat cylinders and tori and some conformally flat manifolds of gen...

Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, $(GR)_n$ structure are completely classified in this short report. A conformally recurrent manifold is shown to be either a de Sitter space or an anti-de space. In particular, spacetime must spacetime.

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also relate the Cotton-York tensor of an umbilic hypersurface to...

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...

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal infinity of a selfdual four-manifolds. We also find a relation between the conformal invariants of the conformal infinity and its ambient.

We solve the σ2-Yamabe problem for a non locally conformally flat manifold of dimension n > 8. Dedicated to Professor W. Y. Ding on the occasion of his 60’s birthday