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Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank of a certain distinguished null parallel distribution $\\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a twofold isometric covering, must be a bundle over the circle with leaves of $\\mathcal{D}^\\perp$ serving as the fibres. The same conclusion holds in the locally homogeneous case if one assumes that $\\,\\mathcal{D}^\\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold, the leaves of $\\mathcal{D}^\\perp$ are the factor manifolds of a global product decomposition.

We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemannian cover if necessary) Kähler–Einstein, provided that λ 2 ≥ - S 12 , where λ 2 is the middle eigenvalue of the self-dual Weyl tensor W + and S is the scalar curvature. An analogous result holds for closed oriented four-manifolds with δ W + = 0 .

We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces. We prove conformal and concircular invariance of some tensors in Eisenhart’s generalized Riemannian spaces. We give new generalizations of symmetric spaces via Eisenhart’s generalized Riemannian spaces. Finally, we describe some properties of covariant derivatives of tensors analogous to Yano’s tensor of concircular curvature in Eisenhart symmetric spaces of various kinds.

We prove the existence of compact pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric, and represent all indefinite metric signatures in all dimensions n ≥ 5 . Until now such manifolds were only known to exist in dimensions n = 3 j + 2 , where j is any positive integer; see Derdzinski and Roter (Ann Global Anal Geom 37(1):73–90, 2010. https://doi.org/10.1007/s10455-009-9173-9 ). As in Derdzinski and Roter (2010), our examples are diffeomorphic to nontrivial torus bundles over the circle and arise from a quotient-manifold construction applied to suitably chosen discrete isometry groups of diffeomorphically-Euclidean “model” manifolds.

In the Alcubierre warp-drive spacetime, we investigate the following scalar curvature invariants: the scalar I , derived from a quadratic contraction of the Weyl tensor, the trace R of the Ricci tensor, and the quadratic r1 and cubic r2 invariants from the trace-adjusted Ricci tensor. In four-dimensional spacetime the trace-adjusted Einstein and Ricci tensors are identical, and their unadjusted traces are oppositely signed yet equal in absolute value. This allows us to express these Ricci invariants using Einstein‘s curvature tensor, facilitating a direct interpretation of the energy-momentum tensor. We present detailed plots illustrating the distribution of these invariants. Our findings underscore the requirement for four distinct layers of an anisotropic stress-energy tensor to create the warp bubble. Additionally, we delve into the Kretschmann quadratic invariant decomposition. We provide a critical analysis of the work by Mattingly et al., particularly their underrepresentation of curvature invariants in their plots by 8 to 16 orders of magnitude. A comparison is made between the spacetime curvature of the Alcubierre warp-drive and that of a Schwarzschild black hole with a mass equivalent to the planet Saturn. The paper addresses potential misconceptions about the Alcubierre warp-drive due to inaccuracies in representing spacetime curvature changes and clarifies the classification of the Alcubierre spacetime, emphasizing its distinction from class B warped product spacetimes.

We study ECS manifolds, that is, pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak, and a rank-one ECS manifold may be called translational or dilational, depending on whether the holonomy group of a natural flat connection in the Olszak distribution is finite or infinite. Some such manifolds are in a natural sense generic, which refers to the algebraic structure of the Weyl tensor. Various examples of compact rank-one ECS manifolds are known: translational ones (both generic and nongeneric) in every dimension n5 , as well as odd-dimensional nongeneric dilational ones, some of which are locally homogeneous. As we show, generic compact rank-one ECS manifolds must be translational or locally homogeneous, provided that they arise as isometric quotients of a specific class of explicitly constructed “model” manifolds. This result is relevant since the clause starting with “provided that” may be dropped: according to a theorem which we prove in another paper, the models just mentioned include the isometry types of the pseudo-Riemannian universal coverings of all generic compact rank-one ECS manifolds. Consequently, all generic compact rank-one ECS manifolds are translational.

This study investigates static charged black holes with Weyl corrections to the Einstein-Maxwell action. By extending the classical solutions, we incorporate the influence of the Weyl tensor coupling with the electromagnetic field tensor, deriving novel black hole solutions under this framework. The characteristics of the derived spacetimes, including their geometrical structure and physical properties, are meticulously examined. Key features explored include geodesics, photon spheres, and quasinormal modes, providing insights into the stability and dynamics around such black holes. Additionally, the deflection of light and the corresponding shadow cast by these black holes are analyzed, revealing the modifications induced by Weyl corrections. This study aims to enrich the understanding of how higher-order tensor interactions can influence observable astrophysical phenomena in the vicinity of compact objects.

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence-free Weyl tensors and -nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider Kähler manifolds with divergence-free Bochner tensors. For quaternion Kähler manifolds, we obtain a partial result towards the LeBrun–Salamon conjecture.

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that CP3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.

Given (M,g0) a closed Riemannian manifold and a nonempty closed subset X in M, the singular σk-Yamabe problem asks for a complete metric g on M\\X conformal to g0 with constant σk-curvature. The σk-curvature is defined as the k-th elementary symmetric function of the eigenvalues of the Schouten tensor of a Riemannian metric. The main goal of this paper is to find solutions to the singular σ2-Yamabe problem with isolated singularities in any nondegenerate closed Riemannian manifold such that the Weyl tensor vanishes to sufficiently high order at the singular points. We will use perturbation techniques and gluing methods.