*-Weyl Curvature Tensor within the Framework of Sasakian and $(\kappa,\mu)$-Contact Manifolds

The object of the present paper is to study $*$-Weyl curvature tensor within the framework of Sasakian and $(\kappa,\mu)$-contact manifolds.

The Weyl Curvature Tensor of Hypersurfaces Under the Mean Curvature Flow

In this paper, we study the evolution of the Weyl curvature tensor W of hypersurfaces in 𝕉n+1 under the mean curvature flow. We find a bound for the Weyl curvature tensor of hypersurfaces during the evolution in terms of time. As a consequence, we suppose that the initial hypersurface is conformally flat, i.e., W =0 at t = 0 and then we find an upper estimate for W during the evolution in terms of time.

On isospectral compactness in conformal class for 4-manifolds

Let [Formula: see text] be a closed 4-manifold with positive Yamabe invariant and with [Formula: see text]-small Weyl curvature tensor. Let [Formula: see text] be any metric in the conformal class of [Formula: see text] whose scalar curvature is [Formula: see text]-close to a constant. We prove that the set of Riemannian metrics in the conformal class [Formula: see text] that are isospectral to [Formula: see text] is compact in the [Formula: see text] topology.

On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.

Tolman mass of spherical fluids with electromagnetic field

Assuming a system with spherical symmetry in f(R) gravity filled with dissipative charged and anisotropic matter, we study the impact of density inhomogeneity and local anisotropy on the gravitational collapse in the presence of charge. For this purpose, we evaluated the modified Maxwell field equations, Weyl curvature tensor, and the mass function. Using Misner–Sharp mass formalism, we construct a relation between the Weyl tensor, density inhomogeneity, and local anisotropy. Specifically, we obtain the expression of modified Tolman mass which helps to analyze the influence of charge and dark source terms on different physical factors, also it helps to study the role of these factors on gravitational collapse.

The Weyl curvature tensor, Cotton–York tensor and gravitational waves: A covariant consideration

[Formula: see text] covariant approach to cosmological perturbation theory often employs the electric part ([Formula: see text]), the magnetic part ([Formula: see text]) of the Weyl tensor or the shear tensor ([Formula: see text]) in a phenomenological description of gravitational waves. The Cotton–York tensor is rarely mentioned in connection with gravitational waves in this approach. This tensor acts as a source for the magnetic part of the Weyl tensor which should not be neglected in studies of gravitational waves in the [Formula: see text] formalism. The tensor is only mentioned in connection with studies of “silent model” but even there the connection with gravitational waves is not exhaustively explored. In this study, we demonstrate that the Cotton–York tensor encodes contributions from both electric and magnetic parts of the Weyl tensor and in directly from the shear tensor. In our opinion, this makes the Cotton–York tensor arguably the natural choice for linear gravitational waves in the [Formula: see text] covariant formalism. The tensor is cumbersome to work with but that should negate its usefulness. It is conceivable that the tensor would equally be useful in the metric approach, although we have not demonstrated this in this study. We contend that the use of only one of the Weyl tensor or the shear tensor, although phenomenologically correct, leads to loss of information. Such information is vital particularly when examining the contribution of gravitational waves to the anisotropy of an almost-Friedmann–Lamitre–Robertson–Walker (FLRW) universe. The recourse to this loss is the use Cotton–York tensor.

CURVATURE STRUCTURE OF SELF-DUAL 4-MANIFOLDS

We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.

CONFORMALLY OSSERMAN MANIFOLDS AND CONFORMALLY COMPLEX SPACE FORMS

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.