We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé’s system of equations. We show that the symmetry group of the Lamé’s system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dependents variables. We obtain the solutions which are invariant under the acti...

A conformally flat hypersurface f:M3→R4 in the four-dimensional Euclidean space R4 is said to be generic if has three distinct principal curvatures everywhere. In this paper, we study hypersurfaces using framework of Möbius geometry. First, classify locally with a vanishing form under transformation group R4. Second, investigate global behavior compact and give some integral formulas about inva...

We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k − 1)-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group c...

The condition for the vanishing of the Weyl tensor is integrated in the spherically symmetric case. Then, the resulting expression is used to find new, conformally flat, interior solutions to Einstein equations for locally anisotropic fluids. The slow evolution of these models is contrasted with the evolution of models with similar energy density or radial pressure distribution but non-vanishin...

Motivated by the prescribing scalar curvature problem, we study the equation ∆gu+Ku p = 0 (1 + ζ ≤ p ≤ n+2 n−2 ) on locally conformally flat manifolds (M, g) with R(g) = 0. We prove that when K satisfies certain conditions and the dimension of M is 3 or 4, any solution u of this equation with bounded energy has uniform upper and lower bounds. Similar techniques can also be applied to prove that...

For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates wou...

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebr...

where C is some constant depending only on (M, g), and k. He also announced in the same paper the same result for general manifolds, without the locally conformally flat assumption. The proof of this claim has not been made available. For general manifolds of dimension n = 3, a proof was given by Li and Zhu in [73]; while for n = 4, a combination of the results of Li and Zhang [70] and Druet [4...

The object of the present paper is to study quasi-conformally flat generalized Sasakian-space-forms. Also we study quasi-conformally semisymmetric generalized Sasakian-space-forms. As a consequence of the results, we obtain some important corollaries.