Explore the vast range of titles available.

In this work, we aim to investigate the characteristics of the Bach and Cotton tensors on Lorentzian manifolds, particularly those admitting a semi-symmetric metric ω-connection. First, we prove that a Lorentzian manifold admitting a semi-symmetric metric ω-connection with a parallel Cotton tensor is quasi-Einstein and Bach flat. Next, we show that any quasi-Einstein Lorentzian manifold admitting a semi-symmetric metric ω-connection is Bach flat.

The aim of this paper is to examine certain properties of Cotton tensor on a para-Kenmotsu manifold of dimension 3 admitting 7-Ricci solitons. The concept of Cotton pseudosymmetric manifolds is also introduced in this paper. Some results based on Cotton flat para-Kenmotsu manifolds that admits m-Ricci solitons are also derived. Furthermore, it is investigated whether some geometric properties of the Cotton tensor do not exist in para-Kenmotsu 3-manifolds. Besides these, we studied Codazzi type of Cotton tensor. Some applications of Cotton tensor in fluid space time are also discussed in the paper.

In this paper, we obtain a classification theorem of real hypersurfaces in a nonflat complex plane whose Cotton tensor is η-vanishing and the structure vector field is an eigenvector field of the Ricci tensor.

In this paper, we study an almost coKähler manifold admitting certain metrics such as ∗ -Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKähler 3-manifold ( M ,  g ) admitting a ∗ -Ricci soliton ( g ,  X ) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKähler ( κ , μ ) -almost coKähler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f . Finally, we prove that a ( κ , μ ) -almost coKähler manifold ( M ,  g ) is coKähler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKähler manifolds which are non-coKähler.

In this paper we find out the evolution formula for the first nonzero eigenvalue of the weighted p-Laplacian operator acting on the space of functions under the Cotton flow on a closed Riemannian 3-manifold M3.

In this paper, we classify three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. We also give some classifications of complete gradient Yamabe solitons with nonpositively curved Ricci curvature in the direction of the gradient of the potential function.

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric inn+2dimensions that encodes a conformal class of metrics inndimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric inn+1dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

In this paper we study on gradient quasi-Einstein solitons with a fourth-order vanishing condition on the Weyl tensor. More precisely, we show that for n ≥ 4, the Cotton tensor of any n -dimensional gradient quasi-Einstein soliton with fourth order f -divergence free Weyl tensor is flat, if the manifold is compact, or noncompact but the potential function satisfies some growth condition. As corollaries, some local characterization results for the quasi-Einstein metrics are derived.

We consider the inverse conductivity problem in a strictly convex domain whose boundary is not known. Usually the numerical reconstruction from the measured current and voltage data is done assuming that the domain has a known fixed geometry. However, in practical applications the geometry of the domain is usually not known. This introduces an error, and effectively changes the problem into an anisotropic one. The main result of this paper is a uniqueness result characterizing the isotropic conductivities on convex domains in terms of measurements done on a different domain, which we call the model domain, up to an affine isometry. As data for the inverse problem, we assume the Robin-to-Neumann map and the contact impedance function on the boundary of the model domain to be given. Also, we present a minimization algorithm based on the use of Cotton-York tensor, which finds the push forward of the isotropic conductivity to our model domain and also finds the boundary of the original domain up to an affine isometry. This algorithm works also in dimensions higher than three, but then the Cotton-York tensor has to replaced with the Weyl tensor.

A bstract We show that a holographic description of four-dimensional asymptotically locally flat spacetimes is reached smoothly from the zero-cosmological-constant limit of anti-de Sitter holography. To this end, we use the derivative expansion of fluid/gravity correspondence. From the boundary perspective, the vanishing of the bulk cosmological constant appears as the zero velocity of light limit. This sets how Carrollian geometry emerges in flat holography. The new boundary data are a two-dimensional spatial surface, identified with the null infinity of the bulk Ricci-flat spacetime, accompanied with a Carrollian time and equipped with a Carrollian structure, plus the dynamical observables of a conformal Carrollian fluid. These are the energy, the viscous stress tensors and the heat currents, whereas the Carrollian geometry is gathered by a two-dimensional spatial metric, a frame connection and a scale factor. The reconstruction of Ricci-flat spacetimes from Carrollian boundary data is conducted with a flat derivative expansion, resummed in a closed form in Eddington-Finkelstein gauge under further integrability conditions inherited from the ancestor anti-de Sitter set-up. These conditions are hinged on a duality relationship among fluid friction tensors and Cotton-like geometric data. We illustrate these results in the case of conformal Carrollian perfect fluids and Robinson-Trautman viscous hydrodynamics. The former are dual to the asymptotically flat Kerr-Taub-NUT family, while the latter leads to the homonymous class of algebraically special Ricci-flat spacetimes.