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The main objective of the present article is to study almost Bach solitons on coKähler manifolds with certain types of potential vector fields. It is shown that the almost Bach solitons reduce to the Bach solitons when the potential vector field is the torse-forming vector field. In this regard, we also prove that the solitons are steady. In the three-dimensional case, considering the potential vector field as concircular, the vector field becomes parallel. Almost gradient Bach solitons are also considered in the present article. Finally, we provide some examples of three-dimensional coKähler manifolds. [ABSTRACT FROM AUTHOR]

We describe all Lorentzian semi-direct extensions of the Heisenberg group which are conformally Einstein. As a by side result, Bach-flat left-invariant Lorentzian metrics on semi-direct extensions of the Heisenberg group are classified, thus providing new background solutions in conformal gravity. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M. Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} with a Batch tensor then it is a Batch flat space-time manifold. [ABSTRACT FROM AUTHOR]

We define a conformally invariant action on gauge connections on a closed pseudo-Riemannian manifold M of dimension 6. At leading order this is quadratic in the gauge connection. The Euler–Lagrange equations of , with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang–Mills equations. For any gauge connection A on M , we define (A) by first defining a Lagrangian density associated to A. This is not conformally invariant but has a conformal transformation analogous to a Q -curvature. Integrating this density provides the conformally invariant action. In the special case that we apply to the conformal Cartan-tractor connection, the functional gradient recovers the natural conformal curvature invariant called the Fefferman–Graham obstruction tensor. So in this case, the Euler–Lagrange equations are exactly the "obstruction-flat" condition for 6-manifolds. This extends known results for 4-dimensional pseudo-Riemannian manifolds where the Bach tensor is recovered in the Yang–Mills equations of the Cartan-tractor connection. [ABSTRACT FROM AUTHOR]

We study a complete connected generalized m-quasi-Einstein manifold M with finite m, admitting a non-homothetic, non-parallel, complete closed conformal vector field V, and show that either M is isometric to a round sphere, or the Ricci tensor can be expressed explicitly in terms of the conformal data over an open dense subset. In the latter case, we prove that M is a warped product of an open real interval with an Einstein manifold; furthermore, it is conformally flat in dimension 4 and has vanishing Cotton and Bach tensors in dimension > 3. Next, we obtain the same explicit expression for the Ricci tensor, and analogous results, for a gradient Ricci almost soliton endowed with a non-parallel closed conformal vector field. [ABSTRACT FROM AUTHOR]

We present a covariant study of static space-times, as such and as solutions of gravity theories. By expressing the relevant tensors through the velocity and the acceleration vectors that characterise static space-times, the field equations provide a natural non-redundant set of scalar equations. The same vectors suggest the form of a Faraday tensor, that is studied in itself and in (non)-linear electrodynamics. In spherical symmetry, we evaluate the explicit expressions of the Ricci, the Weyl, the Cotton and the Bach tensors. Simple restrictions on the coefficients yield well known and new solutions in Einstein, f(R), Cotton and Conformal gravity, with or without charges, in vacuo or with fluid source. [ABSTRACT FROM AUTHOR]

In the present article, our aim is to characterize Bach flat paraSasakian manifolds. It is established that a Bach flat paraSasakian manifold of dimension greater than three is of constant scalar curvature. Next, we prove that if the metric of a Bach flat paraSasakian manifold is a Yamabe soliton, then the soliton field becomes a Killing vector field. Finally, it is shown that a 3-dimensional Bach flat paraSasakian manifold is locally isometric to the hyperbolic space H2n+1(1). [ABSTRACT FROM AUTHOR]

In this paper, we study vacuum static spaces. We firstly derive a Bochner-type formula for the Weyl tensor to vacuum static space. Based on a global argument, under the condition of zero radial Weyl curvature, we then obtain a pointwise identity and use it to prove that each closed vacuum static space of dimension n ≥ 5 with scalar curvature n (n - 1) and zero radial Weyl curvature is Bach flat. [ABSTRACT FROM AUTHOR]

English: In this paper, using special metric deformations introduced by Aubin, we construct Riemannian metrics satisfying non-vanishing conditions concerning the Weyl tensor, on every compact manifold. In particular, in dimension four, we show that there are no topological obstructions for the existence of metrics with non-vanishing Bach tensor. [ABSTRACT FROM AUTHOR]

In this article, we introduce quasi-Bach tensor and correspondingly introduce almost quasi-Bach solitons, thereby generalizing the existing notion of Bach tensor and almost Bach solitons. We explore some properties of gradient quasi Bach solitons with harmonic Weyl curvature tensor. We study the relationships between Weyl tensor, Cotton tensor, tensor D introduced by Cao, and quasi Bach tensor. We also find the evolution of volume, Einstein metric, Ricci curvature and scalar curvature, under the quasi Bach flow. Our results obtained here extends the results of Bach solitons and Bach flow. Finally, we obtain the characterization of gradient quasi-Bach soliton of type I, a particular quasi Bach soliton, on the product manifolds S 2 × H 2 and R 2 × H 2. Our exploration generalizes gradient Bach soliton on R 2 × H 2 obtained by P. T. Ho, while the gradient soliton on S 2 × H 2 is a novel one and is complementary to the results obtained by P. T. Ho. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study K-paracontact manifold admitting parallel Cotton tensor, vanishing Cotton tensor and to study Bach flatness on K-paracontact manifold. In that we prove for a K-paracontact metric manifold M2n+1 has parallel Cotton tensor if and only if M2n+1 is an η-Einstein manifold and r = -2n(2n + 1). Further we show that if g is an n-Einstein K-paracontact metric and if g is Bach flat then g is an Einstein. Also we study vanishing Cotton tensor on (κ,μ)-paracontact manifold for both k > -1 and k < -1. Finally, we prove that if M2n+1 is a (κ,μ)-paracontact manifold for k ≠ -1 and if M2n+1 has vanishing Cotton tensor for κ ≠ μ, then M2n+1 is an η-Einstein manifold. [ABSTRACT FROM AUTHOR]

We introduced and studied Bach almost soliton on Riemannian manifold satisfying certain conditions on the potential vector field. [ABSTRACT FROM AUTHOR]

We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat shrinking gradient solitons are product metrics on R 2 × S 2 and R 2 × H 2 . We also construct a non-Bach-flat expanding homogeneous gradient Bach soliton. We also establish a number of results for solitons to the geometric flow by a general tensor q. [ABSTRACT FROM AUTHOR]

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat. [ABSTRACT FROM AUTHOR]

It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant Q-curvature. This allows us to determine the free data at conformal infinity and to select those choices that lead to Einstein metrics. The asymptotically hyperbolic mass is part of that free data, in contrast to the pure Einstein case. Higher-dimensional generalizations of the Bach tensor lack some of the geometrical meaning of the 4-dimensional case, but for a generalized Bach equation suited to the Fefferman–Graham technique, we are able to obtain a relatively complete result illustrating an interesting splitting of the free data into low-order "Dirichlet" and high-order "Neumann" pairs. [ABSTRACT FROM AUTHOR]

It is known that Einstein metrics are Bach flat. So, the question may arises whether there exists any Riemannian metric which is Bach flat but not Einstein. In this paper, we answer this by constructing several example on certain class of warped product manifold. Indeed the warped product spaces R × f CP n and R × f CH n with the warping function f (t) = k e t (where k is a non zero constant) are non Einstein Bach flat manifolds. These spaces are commonly known as Kenmotsu manifold. Moreover, we prove that a Kenmotsu manifold with parallel Cotton tensor is η -Einstein and Bach flat. Next, we establish that any η -Einstein Kenmotsu manifold of dimension > 3 is Bach flat. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the L 2 -norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kähler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds. [ABSTRACT FROM AUTHOR]

In this paper we study spaces of conformal torsion-free connection of dimension 4 whose connection matrix satisfies the Yang-Mills equations. Here we generalize and strengthen the results obtained by us in previous articles, where the angular metric of these spaces had Minkowski signature. The generalization is that here we investigate the spaces of all possible metric signatures, and the enhancement is due to the fact that additional attention is paid to calculating the curvature matrix and establishing the properties of its components. It is shown that the Yang-Mills equations on 4-manifolds of conformal torsion-free connection for an arbitrary signature of the angular metric are reduced to Einstein's equations, Maxwell's equations and the equality of the Bach tensor of the angular metric and the energy-momentum tensor of the skew-symmetric charge tensor. It is proved that if the Weyl tensor is zero, the Yang-Mills equations have only self-dual or anti-self-dual solutions, i.e the curvature matrix of a conformal connection consists of self-dual or anti-self-dual external 2-forms. With the Minkowski signature (anti)self-dual external 2-forms can only be zero. The components of the curvature matrix are calculated in the case when the angular metric of an arbitrary signature is Einstein, and the connection satisfies the Yang-Mills equations. In the Euclidean and pseudo-Euclidean 4-spaces we give some particular self-dual and anti-self-dual solutions of the Maxwell equations, to which all the Yang-Mills equations are reduced in this case.

Homogeneous conformally Einstein manifolds are classified in dimension four. As a consequence we show that any homogeneous strictly Bach-flat four-dimensional manifold is homothetic to one of the examples constructed by Abbena, Garbiero and Salamon in [1]. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the Bach flow which is defined as ∂ ∂ t g i j = − B i j where B i j is the Bach tensor. Among other things, we study the solitons to the Bach flow. We also study the Bach flow on a four-dimensional Lie group, in which we study the convergence of the Bach flow. [ABSTRACT FROM AUTHOR]

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds. [ABSTRACT FROM AUTHOR]

In this note, we study a gradient generalized m-quasi-Einstein manifold. That is, there exist two smooth functions f, $${\lambda}$$ such that where m is a constant. We first obtain some rigidity results on compact gradient generalized m-quasi-Einstein manifolds. Then, we obtain some classifications for the special gradient generalized m-quasi-Einstein manifolds under the assumption that the Bach tensor is flat. In particular, we obtain some even stronger results in dimension three. [ABSTRACT FROM AUTHOR]

If there exist a smooth function f on $$(M^n, g)$$ and three real constants $$m,\rho ,\lambda $$ ( $$0

Abstract: In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach–Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein–Maxwell equations. Inspired by the work of C. LeBrun on Einstein–Maxwell equations on compact Kähler surfaces, we give a variational characterization of solutions to Bach–Merkulov equations as critical points of the Weyl functional. We also show that extremal Kähler metrics are solutions to these equations, although, contrary to the Einstein–Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces. [Copyright &y& Elsevier]

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S2 × S2, R2 × S2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work. [ABSTRACT FROM AUTHOR]

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite L2 norm in dimension 4. [ABSTRACT FROM AUTHOR]

In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior. [ABSTRACT FROM AUTHOR]

Conformal Einstein spaces are of particular interest in General Relativity and Quantum Gravity. We present a set of necessary and sufficient conditions for a Petrov type N space-time to be conformally related to an empty space. The conditions are developed in two stages: first, we give necessary and sufficient conditions in Newman-Penrose, spinor, and tensor notation for a space to be conformal to a C-space; second, we establish the sufficiency of a set of additional tensorial conditions for a conformal C-space to be conformal to an empty space. [ABSTRACT FROM AUTHOR]

In this work, we study a characterization of gradient Bach solitons (M , g , f) that have a harmonic Weyl curvature, i.e., δ W = 0. Furthermore, we investigate complete 4-dimensional gradient Bach solitons. [ABSTRACT FROM AUTHOR]