Four-dimensional Manifold Research Articles

Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. I. Equations of geodesic lines

Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. I. Equations of geodesic lines

Twistor Bundle of a Neutral Kähler Surface

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

Extrinsic curvature and conformal Gauss–Bonnet for four-manifolds with corner

This paper defines two new extrinsic curvature quantities on the corner of a four-dimensional Riemannian manifold with corner. One of these is a pointwise conformal invariant, and the conformal transformation of the other is governed by a new linear second-order pointwise conformally invariant partial differential operator. The Gauss-Bonnet theorem is then stated in terms of these quantities.

Parallel spinors on globally hyperbolic Lorentzian four-manifolds

We investigate the differential geometry and topology of globally hyperbolic four-manifolds (M, g) admitting a parallel real spinor $$\varepsilon $$ . Using the theory of parabolic pairs recently introduced in [22], we first formulate the parallelicity condition of $$\varepsilon $$ on M as a system of partial differential equations, the parallel spinor flow equations, for a family of polyforms on an appropriate Cauchy surface $$\Sigma \hookrightarrow M$$ . The existence of a parallel spinor on (M, g) induces a system of constraint partial differential equations on $$\Sigma $$ , which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding $$\Sigma \hookrightarrow M$$ . Solutions of this differential system are precisely the allowed initial data for the evolution problem of a parallel spinor and define the notion of parallel Cauchy pair $$({\mathfrak {e}},\Theta )$$ , where $${\mathfrak {e}}$$ is a coframe and $$\Theta $$ is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of Leistner and Lischewski. Furthermore, we classify all compact three-manifolds admitting parallel Cauchy pairs, proving that they are canonically equipped with a locally free action of $${\mathbb {R}}^2$$ and are isomorphic to certain torus bundles over $$S^1$$ , whose Riemannian structure we characterize in detail. Moreover, we classify all left-invariant parallel Cauchy pairs on simply connected Lie groups, specifying when they are allowed initial data for the Ricci flat equations and when the shape operator is Codazzi. Finally, we give a novel geometric interpretation of a class of parallel spinor flows and solve it in several examples, obtaining explicit families of four-dimensional Lorentzian manifolds carrying parallel spinors.

Quantum fluctuations of the metric, three-geometries and conformal embeddings in four-manifolds

In this paper, connections between the path integrals for four-dimensional quantum gravity and string theory are emphasized. It is shown that there is a natural relation between these two path integrals based on the theorems on embeddings of two-dimensional surfaces in four dimensions and four-dimensional manifolds in ten dimensions. The isometry groups of the three-geometries that are spatial hypersurfaces confomally embedded in the four-manifolds are required to be subgroups of [Formula: see text], which is the invariance group of the Pfaffian differential system satisfied by one form in the cotangent bundles on the four-manifolds. Based on this and other physical conditions, the three-geometries are restricted to be [Formula: see text], [Formula: see text] and [Formula: see text] with a boundary, which may be included in the quantum gravitational path integral over four-manifolds which are closed at initial times followed by an exponential expansion compatible with supersymmetry.

Four-dimensional Einstein manifolds with Heisenberg symmetry

We classify Einstein metrics on mathbb {R}^4 invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. We consider metrics which are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-Kähler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.

Postoji li promjena prema B-teoriji vremena?

The purpose of this paper is to explore the connection between change and the B-theory of time, sometimes also called the Scientific view of time, according to which reality is a four-dimensional spacetime manifold, where past, present and future things equally exist, and the present time and non-present times are metaphysically the same. I argue in favour of a novel response to the much-vexed question of whether there is change on the B-theory or not. In fact, B-theorists are often said to hold a ‘static’ view of time. But this far from being innocent label: if the B-theory of time presents a model of temporal reality that is static, then there is no change on the B-theory. From this, one can reasonably think as follows: of course, there is change, so the B-theory must be false. What I plan to do in this paper is to argue that in some sense there is change on the B-theory, but in some other sense, there is no change on the B-theory. To do so, I present three instances of change: Existential Change, namely the view that things change with respect to their existence over time; Qualitative Change, the view that things change with respect to how they are over time; Propositional Change, namely the view that things (i.e. propositions) change with respect to truth value over time. I argue that while there is a reading of these three instances of change that is true on the B-theory, and so there is change on the B-theory in this sense, there is a B-theoretical reading of each of them that is not true on the B-theory, and therefore there is no change on the B-theory in this other sense.

Recurrent curvature over four-dimensional homogeneous manifolds

Recurrent curvature properties are considered on four-dimensional pseudo-Riemannian homogeneous manifolds with non-trivial isotropy, and also on some geometric manifolds.

Gauge theories on compact toric manifolds

We compute the mathcal{N}=2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on mathbb {C}^2. The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the mathbb {C}^2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of mathbb {P}^2 and mathbb {F}_n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a mathcal {N}=2 analog of the mathcal {N}=4 holomorphic anomaly equations.

The mean curvature estimate for the level sets of solutions of the Monge-Ampère equation on Riemannian manifold

For the fully nonlinear elliptic Monge-Ampere equation det D 2 u = 1 with homogeneous Dirichlet boundary value condition, in this paper, a function related to the curvature of the level set of the solution was established, then the differential inequality of the strictly convex solutions of the equation on four-dimensional Riemannian manifold was got. The maximum value of the auxiliary function at the boundary was obtained by using the maximum principle and the mean curvature estimation for the level sets of the solution was given.

SMOOTH FANO 4-FOLDS IN GORENSTEIN FORMATS

AbstractWe construct some new deformation families of four-dimensional Fano manifolds of index one in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the anticanonical embedding in some weighted projective space. They also have relatively smaller anticanonical degree than most other known families of smooth Fano 4-folds.

Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems.

We apply the semi-classical limit of the generalized map for representation of variable-spin systems in a four-dimensional symplectic manifold and approximate their evolution terms of effective classical dynamics on . Using the asymptotic form of the star-product, we manage to “quantize” one of the classical dynamic variables and introduce a discretized version of the Truncated Wigner Approximation (TWA). Two emblematic examples of quantum dynamics (rotor in an external field and two coupled spins) are analyzed, and the results of exact, continuous, and discretized versions of TWA are compared.

Exact Anti-Self-Dual four-manifolds with a Killing symmetry by similarity transformations

We study the group properties and the similarity solutions for the constraint conditions of anti-self-dual null Kähler four-dimensional manifolds with at least a Killing symmetry vector. Specifically we apply the theory of Lie symmetries to determine all the infinitesimal generators of the one-parameter point transformations which leave the system invariant. We use these transformations to define invariant similarity transformations which are used to simplify the differential equations and find the exact form of the spacetime. We show that the constraint equations admit an infinite number of symmetries which can be used to construct an infinite number of similarity transformations.

Untying knots in 4D and Wedderburn’s theorem

It is proved that the Wedderburn Theorem on finite division rings implies that all knots and links in the smooth four-dimensional manifolds are trivial. Communicated by Igor Klep

Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds

Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds

New Cases of Homogeneous Integrable Systems with Dissipation on Tangent Bundles of Four-Dimensional Manifolds

New Cases of Homogeneous Integrable Systems with Dissipation on Tangent Bundles of Four-Dimensional Manifolds

Bach and Einstein’s equations in presence of a field

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four-dimensional smooth manifold [Formula: see text] in the presence of field [Formula: see text], given by a smooth map with source [Formula: see text] and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called [Formula: see text]-Bach tensor, that in absence of the field [Formula: see text] reduces to the classic Bach tensor, and by the vanishing another tensor related to the bi-energy of [Formula: see text]. Since solutions of the Einstein-massless scalar field equations, or more generally, of the Einstein field equations with source the wave map [Formula: see text] solves those generalized Bach’s equations, we include the latter in our analysis providing a systematic study for them, relying on the recent concept of [Formula: see text]-curvatures. We take the opportunity to discuss the related topic of warped product solutions.

A Hopf bifurcation in the Kuramoto-Daido model

A Hopf bifurcation in the Kuramoto-Daido model is investigated based on the generalized spectral theory and the center manifold reduction for a certain class of frequency densities. The dynamical system of the order parameter on a four-dimensional center manifold is derived. It is shown that the dynamical system undergoes a Hopf bifurcation as the coupling strength increases, which proves the existence of a periodic two-cluster state of oscillators.

Analyticity properties of scattering amplitude in theories with compactified space dimensions: The proof of dispersion relations

The analyticity properties of the scattering amplitude for a massive scalar field are reviewed in this paper where the space–time geometry is [Formula: see text], i.e. one spatial dimension is compact. Khuri investigated the analyticity of scattering amplitude in a nonrelativistic potential model in three dimensions with an additional compact dimension. He showed that under certain circumstances, the forward amplitude is nonanalytic. He argued that in high energy scattering if such a behavior persists it would be in conflicts with the established results of quantum field theory and LHC might observe such behaviors. We envisage a real scalar massive field in flat Minkowski space–time in five dimensions. The Kaluza–Klein (KK) compactification is implemented on a circle. The resulting four-dimensional manifold is [Formula: see text]. The LSZ formalism is adopted to study the analyticity of the scattering amplitude. The nonforward dispersion relation is proved. In addition the Jin–Martin bound and an analog of the Froissart–Martin bound are proved. A novel proposal is presented to look for evidence of the large-radius-compactification scenario. A seemingly violation of Froissart–Martin bound at LHC energy might hint that an extra dimension might be decompactified. However, we find no evidence for violation of the bound in our analysis.

Curvature Homogeneous Manifolds in Dimension 4

We classify complete curvature homogeneous metrics on simply connected four-dimensional manifolds which are invariant under a cohomogeneity one action.