Many theories of gravity admit formulations in different, conformally related manifolds, known as the Jordan and Einstein conformal frames. Among them are various scalar-tensor theories of gravity and high-order theories with the Lagrangian f (R) where R is the scalar curvature and f is an arbitrary function. It may happen that a singularity in the Einstein frame corresponds to a regular surfac...

We consider a five-dimensional constant curvature black hole, which is constructed by identifying some points along a Killing vector in a five-dimensional AdS space. The black hole has the topology M4 × S, its exterior is not static and its boundary metric is of the form of a three-dimensional de Sitter space times a circle, which means that the dual conformal field theory resides on a dynamica...

We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analyzed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrat...

We study integral curvature conditions for a Riemannian metric g on S4 that quantify the best bilipschitz constant between (S4,g) and standard S4. Our results show is controlled by L2-norm of Weyl tensor L1-norm Q-curvature, under those quantities are sufficiently small, has positive Yamabe Q-curvature mean-positive. The proof result achieved in two steps. Firstly, we construct quasiconformal m...

The present paper deals with metallic K?hler manifolds. Firstly, we define a tensor H which can be written in terms of the (0,4)-Riemannian curvature and fundamental 2-form manifold study its properties some hybrid tensors. Secondly, weobtain conditions under Hermitian is conformal to manifold. Thirdly, prove that recurrency implies also obtain Riemannian form conformally recurrent non-zero sca...

We derive slow-roll conditions for a scalar field which is non-minimally coupled with gravity in a consistent manner and express spectral indices of scalar/tensor perturbations in terms of the slow-roll parameters. The conformal invariance of the curvature perturbation is proved without linear approximations. Rapid-roll conditions are also derived, and the relation with the slow-roll conditions...

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 with boundary ∂M . We denote the Ricci curvature, scalar curvature, mean curvature, and the second fundamental form by Ric, R , h, and Lαβ , respectively. The Yamabe problem for manifolds with boundary is to find a conformal metric ĝ = eg such that the scalar curvature is constant and the mean curvature is zero. The boundary is call...

We study some properties of a half-lightlike submanifoldM , of a semi-Riemannianmanifold, whose shape operator is conformal to the shape operator of its screen distribution. We show that any screen distribution S(TM) of M is integrable and the geometry of M has a close relation with the nondegenerate geometry of a leaf of S(TM). We prove some results on symmetric induced Ricci tensor and null s...

In this paper we explicitly derive a level set formulation for mean curvature flow in a Riemannian metric space. This extends the traditional geodesic active contour framework which is based on conformal flows. Curve evolution for image segmentation can be posed as a Riemannian evolution process where the induced metric is related to the local structure tensor. Examples on both synthetic and re...