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Weyl's conformal theory of gravity is an extension of Einstein's theory of general relativity which associates metrics with 1-forms . In the case of locally integrable (closed non-exact) 1-forms the spacetime manifolds are no more simply…

We prove that along with the Einstein flow, any small perturbations of an $n(n \geq 4)$-dimensional, non-compact negative Einstein space with some "non-positive Weyl tensor" lead to a unique and global solution, and the solution will be…

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=\lambda g$ for some positive constant $\lambda$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A…

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher…

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the…

We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M,…

By using the York canonical basis of ADM tetrad gravity, in a formulation using radar 4-coordinates for the parametrization of the 3+1 splitting of the space-time, it is possible to write the 4-Riemann tensor of a globally hyperbolic,…

We study various mathematical aspects of the charged rotating black hole with two equal-magnitude angular momenta in five dimensions. We introduce a coordinate system that is regular on the horizon and in which Einstein-Maxwell equations…

We investigate the triviality of compact Ricci solitons under general scalar conditions involving the Weyl tensor. More precisely, we show that a compact Ricci soliton is Einstein if a generic linear combination of divergences of the Weyl…

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes…

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications.…

Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric…

In an n dimensional vector space, any tensor which is antisymmetric in k>n arguments must vanish; this is a trivial consequence of the limited number of dimensions. However, when other possible properties of tensors, for example…

We review recent developments and applications of the classification of the Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the general setup, i.e. main definitions and methods for the classification, some…

The Weyl conformal tensor is the traceless component of the Riemann tensor and therefore, as is known, the information it contains does not appear explicitly in Einstein's equation. Following a rigorous mathematical treatment based on the…

In this work, we study nonconformally Ricci-flat gravitational instantons in four-dimensional Conformal Gravity, both in vacuum and in the presence of nonlinear conformal matter. First, the one-parameter extension of the Kerr-NUT-AdS metric…

We investigate the Weyl tensor algebraic structure of a fully general family of D-dimensional geometries that admit a non-twisting and shear-free null vector field k. From the coordinate components of the curvature tensor we explicitly…

We formulate Eddington's affine gravity in a spacetime which is immersed in a larger eight dimensional space endowed with a hypercomplex structure. The dynamical equation of the first immersed Ricci-type tensor leads to gravitational field…

The algebraic classification of the Weyl tensor in arbitrary dimension n is recovered by means of the principal directions of its "superenergy" tensor. This point of view can be helpful in order to compute the Weyl aligned null directions…

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that "asymmetries cancel". Here we…