We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations oriented two-dimensional manifolds. Our construction relies Regge elements, which are piecewise polynomial symmetric (0,2)-tensor fields possessing single-valued tangential-tangential components along interfaces. When used to discretize Riemannian metric tensor, these tensor do not posse...

On an almost Hermitian manifold, there are two scalar curvatures associated with a canonical connection. In this paper, explicit formulas on these obtained in terms of the Riemannian curvature, norms components covariant derivative fundamental 2-form respect to Levi-Civita connection, and codifferential Lee form. Then we use them get characterization results Kähler metric, balanced locally conf...

This paper is devoted to compute the energy-momentum densities for two exact solutions of the Einstein field equations by using the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and Möller. The spacetimes under consideration are the Weyl-Lewis-Papapetrou and the Levi-Civita metrics. The Weyl metric becomes the special case of the Weyl-Lewis-Papapetrou solution. The Levi-Civita metric p...

We obtain the most general static cylindrically symmetric vacuum solutions of the Einstein field equations in (4+N) dimensions. Under the assumption of separation of variables, we construct a family of Levi-Civita-Kasner vacuum solutions in (4 + N). We discuss the dimensional reduction of the static solutions. Depending on the reduction procedure, they can be interpreted either as a scalar-vacu...

We present the whole set of equations with regularity and matching conditions required for the description of physically meaningful static cylindrically symmmetric distributions of matter, smoothly matched to Levi-Civita vacuum spacetime. It is shown that the conformally flat solution with equal principal stresses represents an incompressible fluid. It is also proved that any conformally flat c...

Let Fm = (M,F ) be a Finsler manifold and G be the Sasaki– Finsler metric on the slit tangent bundle TM0 = TM {0} of M . We express the scalar curvature ρ̃ of the Riemannian manifold (TM0, G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ρ̃ to be a positively homogenenous function of degree zero with respect to the fiber coo...

If ∇ is a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl’s result that W vanishes if and only if (M,∇) is (locally) diffeomorphic to RP with ∇, when transported to RP, in the projective class of ∇LC , the Levi-Civita connection of the Fubini–Study metric on...