Nonzero Scalar Curvature Research Articles

A quadric ansatz method for a certain class of second order PDEs

We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional Kähler-Einstein equation with symmetry and the one-sided type-D Einstein equation with nonzero scalar curvature. The procedure, which reduces the PDEs to ordinary differential equations (ODEs), involves imposing additional constraints to the form of the ansatz, depending on the exact form of the PDEs. Thus its applicability varies within the class. We identify applicable subclasses of a class of modified Toda equations, one of which includes the Kähler-Einstein equation with symmetry, and give the corresponding reduced ODEs. In addition, we obtain a quadric ansatz reduction for a family of Einstein-Weyl equations in arbitrary dimensions. Lastly, we suggest a particular form of equations for which our procedure appears to be most effective as the aforementioned constraints vanish, and also present some reduction results using another type of ansatz, namely the hyperplane ansatz.

Ricci Vector Fields

We introduce a special vector field ω on a Riemannian manifold (Nm,g), such that the Lie derivative of the metric g with respect to ω is equal to ρRic, where Ric is the Ricci tensor of (Nm,g) and ρ is a smooth function on Nm. We call this vector field a ρ-Ricci vector field. We use the ρ-Ricci vector field on a Riemannian manifold (Nm,g) and find two characterizations of the m-sphere Smα. In the first result, we show that an m-dimensional compact and connected Riemannian manifold (Nm,g) with nonzero scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nonconstant function and the integral of Ricω,ω has a suitable lower bound that is necessary and sufficient for (Nm,g) to be isometric to m-sphere Smα. In the second result, we show that an m-dimensional complete and simply connected Riemannian manifold (Nm,g) of positive scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of ω has an appropriate upper bound, if and only if (Nm,g) is isometric to m-sphere Smα.

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

Gravitational potential in spherical topologies

We study the properties of the Newtonian gravitational potential in a spherical Universe for different topologies. For this, we use the non-Euclidean Newtonian theory developed in Vigneron [On non-Euclidean Newtonian theories and their cosmological backreaction, Classical Quantum Gravity 39, 155006 (2022)]. describing Newtonian gravitation in a spherical or hyperbolic Universe. The potential is calculated for a point mass in all the globally homogeneous regular spherical topologies, i.e., whose fundamental domain is unique and is a platonic solid. We provide the exact solution and the Taylor expansion series of the potential at a test position near the point mass. We show that the odd terms of the expansion can be interpreted as coming from the presence of a nonzero spatial scalar curvature, while the even terms relate to the closed nature of the topological space. A consequence is that, compared to the point mass solution in a 3-torus, widely used in Newtonian cosmological simulations, the spherical cases all feature an additional attractive first order term dependent solely on the spatial curvature. The choice of topology only affects the potential at second order and higher. For typical estimates of cosmological scales (curvature and topology), the strongest topological effect occurs in the case of the Poincar\'e dodecahedral space, but in general the effect of curvature dominates over topology. We also provide the set of equations that can be used to perform $N$-body simulations of structure formation in spherical topologies.

On concurrent vector fields on Riemannian manifolds

<abstract><p>It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.</p></abstract>

A STUDY ON MANIFOLD KAEHLERIAIN SPACE

This work delves into the space-time theory of the 4-dimensional Kaehler manifold. Since the isotropic pressure, energy density, and energy momentum tensor all vanish in a perfect fluid Kaehler space-time manifold, we have established that this space-time manifold is an Einstein manifold and studied the Einstein equation with a cosmological constant in it. Finally, we demonstrated that, on a conformally flat, perfectly fluid Kaehler space-time manifold, the velocity vector field is infinitesimally spatially isotropic. Ideal fluid dilution to the point where only Ricci and minimal symmetry hold. We have proven that Kaehler space-time manifolds have either 0 scalar curvature or a connection between the respective rho and alpha vector fields via g(rho,alpha) = 4. To conclude, we have proved that a Kaehler space-time manifold cannot have both perfect fluidity and non-zero scalar curvature (weak Ricci symmetry).

Geometric theory of Weyl structures

Given a parabolic geometry on a smooth manifold [Formula: see text], we study a natural affine bundle [Formula: see text], whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on [Formula: see text], which induces an almost bi-Lagrangian structure on [Formula: see text] and a compatible linear connection on [Formula: see text]. We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with nonzero scalar curvature, provided that the parabolic geometry is torsion-free and [Formula: see text]-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in [Formula: see text]. For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge–Ampere equation and thus to properly convex projective structures.

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

We answer in the affirmative the question posed by Conti and Rossi [7,8] on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.

Yang-Mills model for centrally extended 2D gravity

A Yang-Mills theory linear in the scalar curvature for two-dimensional gravity with symmetry generated by the semidirect product formed with the Lie derivative of the algebra of diffeomorphisms with the two-dimensional Abelian algebra is formulated. As compared with dilaton models, the role of the dilaton is played by the dual field strength of a $U(1)$ gauge field. All vacuum solutions are found. They are either black holes or have constant scalar curvature. Those with constant scalar curvature have constant dual field strength. In particular, solutions with vanishing cosmological constant but nonzero scalar curvature exist. In the conformal-Lorenz gauge, the model has a conformal field theory interpretation whose residual symmetry combines holomorphic diffeomorphisms with a subclass of $U(1)$ gauge transformations while preserving two-dimensional de Sitter and anti-de Sitter boundary conditions. This is the same symmetry as in Jackiw-Teitelboim-Maxwell gravity considered by Hartman and Strominger. It is argued that this is the only nontrivial Yang-Mills model linear in the scalar curvature that exists for real Lie algebras of dimension four.

Para-K\xe4hler-Einstein 4-manifolds and non-integrable twistor distributions

We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

Some Results on Ricci Almost Solitons

We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r2≤nσr on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue λ satisfying n2λ2≥r2; the third result uses a suitable lower bound on the Ricci curvature S(w,w). Finally, we show that an n-dimensional connected Ricci almost soliton (M,g,w,σ) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, nσ−r is a constant along integral curves of w and the Ricci curvature S(w,w) has a suitable lower bound.

Some notes on metallic Kähler manifolds

The present paper deals with metallic K?hler manifolds. Firstly, we define a tensor H which can be written in terms of the (0,4)-Riemannian curvature tensor and the fundamental 2-form of a metallic K?hler manifold and study its properties and some hybrid tensors. Secondly, weobtain the conditions under which a metallic Hermitian manifold is conformal to a metallic K?hler manifold. Thirdly, we prove that the conformal recurrency of a metallic K?hler manifold implies its recurrency and also obtain the Riemannian curvature tensor form of a conformally recurrent metallic K?hler manifold with non-zero scalar curvature. Finally, we present a result related to the notion of Z recurrent form on a metallic K?hler manifold.

Indefinite Einstein metrics on nice Lie groups

Abstract We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension ≥ 8 {\geq 8} .

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

Holonomy algebras of Einstein pseudo-Riemannian manifolds

The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the para-quaternionic-K\"ahlerian manifolds with non-zero scalar curvature are considered. Einstein not Ricci-flat metrics of signature $(2,n)$ with all possible holonomy algebras are given.

Gauge Theory on Projective Surfaces and Anti-self-dual Einstein Metrics in Dimension Four

Given a projective structure on a surface N, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space M of a certain rank 2 affine bundle M rightarrow N. The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on mathbb {RP}^2 is the non-compact real form of the Fubini–Study metric on M=mathrm {SL}(3, mathbb {R})/mathrm {GL}(2, mathbb {R}). We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.

Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces

The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $\omega$. This article studies limits of manifolds for which $\omega$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperk\"ahler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperk\"ahler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.

KERR–NEWMAN BLACK HOLES IN f(R) THEORIES

In the context of f(R) modified gravity theories, we study the Kerr–Newman black hole solutions. We study nonzero constant scalar curvature solutions and discuss the metric tensor that satisfies the modified field equations. We conclude that, in the absence of a cosmological constant, the black holes (BHs) existence is determined by the sign of a parameter h dependent of the mass, the charge, the spin and the scalar curvature. Different values of this parameter lead to diverse astrophysical objects, such as extremal and marginal extremal BHs. Thermodynamics of BHs are then studied, as well as their local and global stability. We analyze these features in a large variety of f(R) models. We remark the main differences with respect to general relativity and show the rich thermodynamical phenomenology that characterizes this framework.

A note on renormalized volume functionals

New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compactification. The second variation of renormalized volume functionals under conformal change is identified, and is used to show that Einstein metrics of nonzero scalar curvature are local extrema.

SOME ASPECTS OF SPHERICAL SYMMETRIC EXTREMAL DYONIC BLACK HOLES IN 4D N = 1 SUPERGRAVITY

In this paper, we study several aspects of extremal spherical symmetric black hole solutions of four-dimensional N = 1 supergravity coupled to vector and chiral multiplets with the scalar potential turned on. In the asymptotic region, the complex scalars are fixed and regular which can be viewed as the critical points of the black hole and the scalar potentials with vanishing scalar charges. It follows that the asymptotic geometries are of a constant and nonzero scalar curvature which are generally not Einstein. These spaces could also correspond to the near horizon geometries which are the product spaces of a two anti-de Sitter surface and the two sphere if the value of the scalars in both regions coincide. In addition, we prove the local existence of nontrivial radius dependent complex scalar fields which interpolate between the horizon and the asymptotic region. We finally give some simple ℂn-models with both linear superpotential and gauge couplings.