In [Hz], Heinz proved that there is no harmonic diffeomorphism from the unit disk D onto the complex plane C. The result was generalized by Schoen [S] and he proved that there is no harmonic diffeomorphism from the unit disk onto a complete surface of nonnegative curvature. Unlike conformal or quasi-conformal maps between Riemann surfaces, the inverse of a harmonic map is not harmonic in genera...

In this proceeding, we review modified theories of gravity with a curvature-matter coupling between an arbitrary function the scalar curvature and Lagrangian density matter. This explicit nonminimal induces non-vanishing covariant derivative energy-momentum tensor, that implies non-geodesic motion consequently leads to appearance extra force. Here, explore physical cosmological implications non...

We present some results in [9], a continuation of our earlier works [7] and [8]. One result is the existence and compactness of solutions to a fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds, and the other is a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations. Let (M, g) be an n−dimensional, ...

The diagonalization of the metrical and canonical Hamilton operators of a scalar field with an arbitrary coupling, with a curvature in N-dimensional homogeneous isotropic space is considered in this paper. The energy spectrum of the corresponding quasiparticles is obtained and then the modified energy-momentum tensor is constructed; the latter coincides with the metrical energy-momentum tensor ...

When Einstein was thinking about the theory of general relativity based on the elimination of especial relativity constraints (especially the geometric relationship of space and time), he understood the first limitation of especial relativity is ignoring changes over time. Because in especial relativity, only the curvature of the space was considered. Therefore, tensor calculations should be to...

This article presents a new definition of Branson’s Q-curvature in even dimensional conformal geometry. The Q-curvature is a generalization of the scalar curvature in dimension 2: it satisfies an analogous transformation law under conformal rescalings of the metric and on conformally flat manifolds its integral is a multiple of the Euler characteristic. Our approach is motivated by the recent w...

The aim of the present paper is to provide a global presentation, as complete as we can, of the theory of special Finsler manifolds. We introduce and investigate globally (or intrinsically, free from local coordinates) many of the most important and commonly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg, general Landesberg, P -reducible, C-reducible, semi-C-reducible...

We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our fl...

Static spherically symmetric solutions for conformal gravity in three dimensions are found. Black holes and wormholes are included within this class. Asymptotically the black holes are spacetimes of arbitrary constant curvature, and they are conformally related to the matching of different solutions of constant curvature by means of an improper conformal transformation. The wormholes can be con...