In this paper, we prove the following two results: First, we study a class of conformally invariant operators P and their related conformally invariant curvatures Q on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, Q-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaf...

In this article, we discuss the global aspects of geometry locally conformally flat (complete and compact) Riemannian manifolds. particular, article reviews improves some results (e.g., conditions compactness degeneration into spherical or space forms) on “in large" The presented here were obtained using generalized classical Bochner technique, as well Ricci flow.

It is a classical fact that the cotangent bundle T M of a differentiable manifold M enjoys a canonical symplectic form Ω. If (M, J, g, ω) is a pseudo-Kähler or para-Kähler 2n-dimensional manifold, we prove that the tangent bundle TM also enjoys a natural pseudo-Kähler or para-Kähler structure (J̃, g̃,Ω), where Ω is the pull-back by g of Ω and g̃ is a pseudoRiemannian metric with neutral signature ...

This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect...

The Newman-Penrose-Perjes formalism is applied to Sasakian 3-manifolds and the local form of the metric and contact structure is presented. The local moduli space can be parameterised by a single function of two variables and it is shown that, given any smooth function of two variables, there exists locally a Sasakian structure with scalar curvature equal to this function. The case where the sc...

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive. Dedicated to the memory of Profess...

The geometric properties of the di-meron solution to the SU (2) Yang-Mills equations are studied in detail. The essential geometric structure of this solution is that of a locally symmetric space endowed with a Riemannian structure which is conformally flat. The di-meron solution is representable by an integrable 3-distribution over Euclidean 4-space. The corresponding integral surfaces are obt...

In quantum mechanics the kinetic energy term for a single particle is usually written in the form of the Laplace-Beltrami operator. This operator is a factor ordering of the classical kinetic energy. We investigate other relatively simple factor orderings and show that the only other solution for a conformally flat metric is the conformally invariant Laplace-Beltrami operator. For non-conformal...