Four-dimensional Manifold Research Articles

Test of emergent gravity

In this paper we examine a small but detailed test of the emergent gravity picture with explicit solutions in gravity and gauge theory. We first derive symplectic U(1) gauge fields starting from the Eguchi-Hanson metric in four-dimensional Euclidean gravity. The result precisely reproduces the U(1) gauge fields of the Nekrasov-Schwarz instanton previously derived from the top-down approach. In order to clarify the role of noncommutative spacetime, we take the Braden-Nekrasov U(1) instanton defined in ordinary commutative spacetime and derive a corresponding gravitational metric. We show that the K\"ahler manifold determined by the Braden-Nekrasov instanton exhibits a spacetime singularity while the Nekrasov-Schwarz instanton gives rise to a regular geometry-the Eguchi-Hanson space. This result implies that the noncommutativity of spacetime plays an important role for the resolution of spacetime singularities in general relativity. We also discuss how the topological invariants associated with noncommutative U(1) instantons are related to those of emergent four-dimensional Riemannian manifolds according to the emergent gravity picture.

Stationary model of the Universe with torsion

On a four-dimensional pseudo-Riemannian manifold with the metric of a stationary model of the Universe, we construct a Riemann-Cartan structure with the automorphism group of maximum dimension. The torsion tensor of this structure is the sum of two parts: semisymmetric, aspiring to geometrization of the spin density of matter, and skew-symmetric, determining the torsion of a spatial section. We give a geometric interpretation of the spatial section torsion. We prove that the maximum dimension of the Lie group of automorphisms of a Riemann-Cartan space-time manifold with a semisymmetric or skewsymmetric connection is seven.

Locally Homogeneous Four-Dimensional Manifolds of Signature (2,2)

In this paper we consider 4-dimensional neutral-signature curvature models and obtain the complete classification of the Ricci operator. We then consider the property of curvature homogeneity for the above manifolds and prove that every complete, connected and simply connected 1-curvature homogeneous 4-dimensional manifold of signature (2,2) with a non-degenerate Ricci operator is isometric to a four-dimensional Lie group equipped with a left invariant neutral metric. We also classify Ricci-parallel curvature homogeneous 4-dimensional manifolds of signature (2,2).

Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension

Let $$\mathrm{M }^n,\, n \in \{4,5,6\}$$ , be a compact, simply connected $$n$$ -manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $$\mathrm{M }^n$$ by a torus $$\mathrm{T }^{n-2}$$ is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

Computation of partially invariant solutions for the Einstein Walker manifolds’ identifying equations

In this paper, partially invariant solutions (PISs) method is applied in order to obtain new four-dimensional Einstein Walker manifolds. This method is based on subgroup classification for the symmetry group of partial differential equations (PDEs) and can be regarded as the generalization of the similarity reduction method. For this purpose, those cases of PISs which have the defect structure δ=1 and are resulted from two-dimensional subalgebras are considered in the present paper. Also it is shown that the obtained PISs are distinct from the invariant solutions that obtained by similarity reduction method.

A Generalization of the Goldberg–Sachs theorem and its consequences

The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi-Yau or symplectic and admits a solution for the source-free Einstein-Maxwell equations.

Triple M-brane configurations and preserved supersymmetries

We investigate all standard triple composite M-brane intersections defined on products of Ricci-flat manifolds for preserving supersymmetries in eleven-dimensional N=1 supergravity. The explicit formulae for computing the numbers of preserved supersymmetries are obtained, which generalize the relations for topologically trivial flat factor spaces presented in the classification by Bergshoeff et al. We obtain certain examples of configurations preserving some fractions of supersymmetries, e.g. containing such factor spaces as K3, C⁎2/Z2, a four-dimensional pp-wave manifold and the two-dimensional pseudo-Euclidean manifold R⁎1,1/Z2.

ON THE EQUATIONS DETERMINING THE RICCI FLOWS ON MANIFOLDS

The examples of the Ricci flows on the four-dimensional manifolds defined by solutions of nonlinear evolution differential equations of the type Monge–Ampere constructed. Their particular solutions are derived and their properties are discussed.

Why time flows

The question used as the title of this article has been asked frequently since the introduction of space–time in which both space and time function as coordinates in a four-dimensional manifold. After a brief review of space–time geometry using the affine bundle it is shown how to construct it from a bundle of linear frames of a five-dimensional manifold. The flow of time is a consequence of the construction.

Weyl tensor classification in four-dimensional manifolds of all signatures

It is well known that the classification of the Weyl tensor in Lorentzian manifolds of dimension four, the so called Petrov classification, was a great tool to the development of general relativity. Using the bivector approach it is shown in this article a classification for the Weyl tensor in all four-dimensional manifolds, including all signatures and the complex case, in an unified and simple way. The important Petrov classification then emerges just as a particular case in this scheme. The boost weight classification is also extended here to all signatures as well to complex manifolds. For the Weyl tensor in four dimensions it is established that this last approach produces a classification equivalent to the one generated by the bivector method.

Harmonic morphisms and shear-free perfect fluids coupled with gravity

We employ the technique used in the classification of harmonic morphisms with one-dimensional fibers on four-dimensional Einstein manifolds (Pantilie in Commun Anal Geom 10:779–814, 2002) to give a simpler proof of the fact that the shear-free perfect fluids coupled to gravity are either irrotational or expansion free in the case when the equation of state is $$\rho =-3p$$ .

A note on supersymmetric AdS6 solutions of massive type IIA supergravity

Motivated by the AdS_6/CFT_5 correspondence, we study general supersymmetric solutions of massive type IIA supergravity, consisting of a warped product of six-dimensional anti-de Sitter space AdS_6 with a four-dimensional Riemannian manifold M_4, and fluxes compatible with the SO(5,2) symmetry. We find that the only local solution of this form is the warped AdS_6 x_w S^4 solution, originally discovered by Brandhuber and Oz. We discuss the supersymmetric properties of this solution.

COMPLEX AND PARACOMPLEX STRUCTURES ON HOMOGENEOUS PSEUDO-RIEMANNIAN FOUR-MANIFOLDS

We completely classify invariant Hermitian and Kähler structures, together with their paracomplex analogues, on four-dimensional homogeneous pseudo-Riemannian manifolds with nontrivial isotropy subalgebra.

Some General New Einstein Walker Manifolds

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

Real AlphaBeta-geometries and Walker geometry

By a real αβ-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of α-planes and an integrable distribution of β-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i.e., is a Walker geometry) and the three-dimensional distribution spanned by the α- and β-distributions is integrable. The case when both distributions are parallel, which has been called two-sided Walker geometry, is obtained as a special case. We also study real αβ-geometries for which the corresponding spinors are both multiple Weyl principal spinors. All these results have natural analogues in the context of the hyperheavens of complex general relativity.

Integrable vortex-type equations on the two-sphere

We consider the Yang-Mills instanton equations on the four-dimensional manifold S^2xSigma, where Sigma is a compact Riemann surface of genus g>1 or its covering space H^2=SU(1,1)/U(1). Introducing a natural ansatz for the gauge potential, we reduce the instanton equations on S^2xSigma to vortex-type equations on the sphere S^2. It is shown that when the scalar curvature of the manifold S^2xSigma vanishes, the vortex-type equations are integrable, i.e. can be obtained as compatibility conditions of two linear equations (Lax pair) which are written down explicitly. Thus, the standard methods of integrable systems can be applied for constructing their solutions. However, even if the scalar curvature of S^2xSigma does not vanish, the vortex equations are well defined and have solutions for any values of the topological charge N. We show that any solution to the vortex equations on S^2 with a fixed topological charge N corresponds to a Yang-Mills instanton on S^2xSigma of charge (g-1)N.

On Euler’s stretching formula in continuum mechanics

The Lie time-derivative of the material metric tensor field along the motion is the proper mathematical definition of the physical notion of strain rate or stretching. Its expression, as symmetric part of the velocity gradient in Euclid space, is provided by a celebrated formula conceived by the genius of Leonhard Euler around the middle of the eighteenth century and since then reproduced in articles, books and treatises on continuum mechanics. We present here a formulation, in the proper geometric context of the four-dimensional space-time manifold endowed with an arbitrary linear connection and referring to a material body of arbitrary dimensionality. The expression involves the material time-derivative of the metric field and torsion-form and gradient of the velocity field, according to the connection induced on the trajectory. As an application, the expressions of the Gram matrix of the stretching in natural and in normalized (or engineering) reference systems induced by orthogonal polar coordinates are provided.

A note on four-dimensional (anti-)self-dual quasi-Einstein manifolds

In this short note we prove that any complete four-dimensional anti-self-dual (or self-dual) quasi-Einstein manifold is either Einstein or locally conformally flat. This generalizes a recent result of X. Chen and Y. Wang.

On the blow-up of four-dimensional Ricci flow singularities

Abstract. In this paper we prove a conjecture by Feldman–Ilmanen–Knopf (2003) that the gradient shrinking soliton metric they constructed on the tautological line bundle over ℂℙ 1 $\mathbb {CP}^1$ is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have non-negative Ricci curvature.

Equivalence between the Osserman condition and the Rakić duality principle in dimension 4

We show that four-dimensional Riemannian manifolds which satisfy the Rakić duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant). Thus, since it was proved in Rakić (1999) [9] that Osserman manifolds satisfy the Rakić duality principle, both conditions are equivalent.