Four-dimensional Manifold Research Articles

New cases of integrable ninth-order conservative and dissipative dynamical systems

New cases of integrable dynamical systems of the ninth order homogeneous in terms of variables are presented, in which a system on a tangent bundle to a four-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of a different sign. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

Kirby Diagram of Polar Flows on Four-Dimensional Manifolds

Kirby Diagram of Polar Flows on Four-Dimensional Manifolds

Boundary Liouville conformal field theory in four dimensions

Higher dimensional Euclidean Liouville conformal field theories (LCFTs) consist of a log-correlated real scalar field with a background charge and an exponential potential. We analyse the LCFT on a four-dimensional manifold with a boundary. We extend to the boundary, the conformally covariant GJMS operator and the Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{Q} $$\\end{document}-curvature term in the LCFT action and classify the conformal boundary conditions. Working on a flat space with plate boundary, we calculate the dimensions of the boundary conformal primary operators, the two- and three-point functions of the displacement operator and the boundary conformal anomaly coefficients.

Unification of conformal gravity and internal interactions

Based on the observation that the dimension of the tangent space is not necessarily equal to the dimension of the corresponding curved manifold and on the known fact that gravitational theories can be formulated in a gauge theoretic way, we discuss how to describe all known interactions in a unified manner. This is achieved by enlarging the tangent group of the four-dimensional manifold to SO(2, 16), which permits the inclusion of both gauge groups, the one that describes gravity as a gauge theory as well as the SO(10) describing the internal interactions. Moreover it permits the use of both Weyl and Majorana conditions imposed on the fermions, as to avoid the duplication of fermion multiplets of SO(10) appearing in previous attempts. The gravity theory discussed in the present work is the Conformal Gravity which, after a spontaneous symmetry breaking, can lead either to Weyl Gravity or to the usual Einstein Gravity.

Primordial Black Holes from Spatially Varying Cosmological Constant Induced by Field Fluctuations in Extra Dimensions

The origin and evolution of supermassive black holes (SMBHs) in our universe have sparked controversy. In this study, we explore the hypothesis that some of these black holes may have seeded from the direct collapse of dark energy domains with density significantly higher than the surrounding regions. The mechanism of the origin of such domains relies on the inflationary evolution of a scalar field acting in D dimensions, which is associated with the cosmological constant in our four-dimensional spacetime manifold. Inner space quantum fluctuations of the field during inflation are responsible for the spatial variations of the dark energy density in our space. This finding holds particular significance, especially considering recent evidence from pulsar timing array observations, which supports the existence of a stochastic gravitational wave background consisting of SMBH mergers.

The Kastler–Kalau–Walze-type theorems about J-Witten deformation

In this paper, we obtain a Lichnerowicz-type formula for [Formula: see text]-Witten deformation and give the proof of the Kastler–Kalau–Walze-type theorems associated with [Formula: see text]-Witten deformation on four-dimensional and six-dimensional almost product Riemannian manifold with (respectively, without) boundary. We give an explanation of the Einstein–Hilbert action for [Formula: see text]-Witten deformation on four-dimensional manifold with boundary.

Double field theory, twistors, and integrability in 4-manifolds

The search for a geometrical understanding of dualities in string theory, in particular T-duality, has led to the development of modern T-duality covariant frameworks such as Double Field Theory, whose mathematical structure can be understood in terms of generalized geometry and, more recently, para-Hermitian geometry. In this work we apply techniques associated to this doubled geometry to four-dimensional manifolds, and we show that they are particularly well-suited to the analysis of integrability in special spacetimes, especially in connection with Penrose's twistor theory and its applications to general relativity. This shows a close relationship between some of the geometrical structures in the para-Hermitian approach to double field theory and those in algebraically special solutions to the Einstein equations. Particular results include the classification of four-dimensional, possibly complex-valued, (para-)Hermitian structures in different signatures, the Lie and Courant algebroid structures of special spacetimes, and the analysis of deformations of (para-)complex structures. We also discuss a notion of “weighted algebroids” in relation to a natural gauge freedom in the framework. Finally, we analyze the connection with two- and three-dimensional (real and complex) twistor spaces, and how the former can be understood in terms of the latter, in particular in terms of twistor families.

Tangent Bundles Endowed with Quarter-Symmetric Non-Metric Connection (QSNMC) in a Lorentzian Para-Sasakian Manifold

The purpose of the present paper is to study the complete lifts of a QSNMC from an LP-Sasakian manifold to its tangent bundle. The lifts of the curvature tensor, Ricci tensor, projective Ricci tensor, and lifts of Einstein manifold endowed with QSNMC in an LP-Sasakian manifold to its tangent bundle are investigated. Necessary and sufficient conditions for the lifts of the Ricci tensor to be symmetric and skew-symmetric and the lifts of the projective Ricci tensor to be skew-symmetric in the tangent bundle are given. An example of complete lifts of four-dimensional LP-Sasakian manifolds in the tangent bundle is shown.

Time dilation and the equivalence of inertial frames

It is well known that simultaneity within an inertial frame is defined in relativity theory by a convention or definition. This definition leads to different simultaneities across inertial frames and the well-known principle of relativity of simultaneity. The lack of a universal present implies the existence of past, present, and future as a collection of events on a four-dimensional manifold or continuum wherein three dimensions are space like and one dimension is time like. However, such a continuum precludes the possibility of evolution of future from the present as all events exist “forever” so to speak on the continuum with the tenses past, present, and future merely being perceptions of different inertial frames. Such a far-reaching ontological concept, created by a mere convention, is yet to gain full acceptance. In this paper, we present arguments in favor of an absolute present, which means simultaneous events are simultaneous in all inertial frames, and subscribe to evolution of future from the present.

Weakly-Einstein conditions over four-dimensional homogenous pseudo-Riemannian manifolds

A curvature identity is studied through homogenous pseudo-Riemannian four-manifolds with nontrivial isotropy. Then, special classes of the considered spaces i.e. weakly-Einstein examples are completely classified.

Conformal perturbations of modified Novikov operators and the Kastler–Kalau–Walze type theorem

In this paper, we establish two Kastler–Kalau–Walze type theorems for conformal perturbations of modified Novikov operators on four-dimensional and six-dimensional compact manifolds with (respectively, without) boundary.

Second Chern-Einstein metrics on four-dimensional almost-Hermitian manifolds

Abstract We study four-dimensional second Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis, the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe four-dimensional compact second Chern-Einstein locally conformally symplectic manifolds, and we give some examples of such manifolds. Finally, we study the second Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second Chern-Einstein metric with a parallel non-zero Lee form.

Four-dimensional Riemannian product manifolds with circulant structures

"A 4-dimensional Riemannian manifold equipped with an additional tensor structure, whose fourth power is the identity, is considered. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant, and it acts as an isometry with respect to the metric. The Riemannian product manifold associated with the considered manifold is studied. Conditions for the metric, which imply that the Riemannian product manifold belongs to each of the basic classes of Staikova-Gribachev's classi cation, are obtained. Examples of such manifolds are given."

On an Indefinite Metric on a Four-Dimensional Riemannian Manifold

Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type (1,1), whose fourth power is minus the identity. The additional structure is skew-circulant and compatible with the metric, such that an isometry is induced in every tangent space on the manifold. Both structures define an indefinite metric. With the help of the indefinite metric, we determine circles in different two-planes in the tangent space on the manifold. We also calculate the length and area of the circles. On a smooth closed curve, such as a circle, we define a vector force field. Further, we obtain the circulation of the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analog of the well-known Green’s formula in the Euclidean space.

The Open Universe: Totality, Self-reference and Time

ABSTRACT Before the twentieth century, the Universe was usually imagined as a large spatially extended thing unfolding in time. The past was fixed and the future was open; unfolding was conceived as an asymmetric process of coming into being. Relativity introduced a new vision in which space and time are presented together as a single four-dimensional manifold of events. That, together with the fact that the fundamental laws of our classical theories are symmetric in time, made understanding why the past and future present themselves so differently in our experience one of the central challenges of physics. The last two centuries have seen a great deal of progress in understanding various of the so-called arrows of time: the thermodynamic arrow, the cosmological arrow, the arrow of knowledge or information. There remains an outstanding piece of this puzzle that has seen little progress, one that Roger Penrose described in his beautiful paper Singularities and Time-Asymmetry in 1979 as: “The arrow most difficult to comprehend … namely the feeling of relentless forward temporal progression, according to which potentialities seem to be transformed into actualities.” I will propose that the insight needed to resolve the problem involves taking into account that we are part of the universe and that any attempt to model it as a totality involves self-reference. I will argue specifically that self-reference, against the background of a thermodynamic gradient, creates an instability in an embedded agent’s ability to know the future or even treat it as a potential object of knowledge. That instability captures the sense in which the future remains for her perpetually open and the passage of time resolves openness into the fixity of fact.

Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that mathbb {C}mathbb {P}^3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.

Stability and energy identity for Yang–Mills–Higgs pairs

In this paper, we study the properties of the critical points of Yang–Mills–Higgs functional, which are called Yang–Mills–Higgs pairs. We first consider the properties of weakly stable Yang–Mills–Higgs pairs on a vector bundle over Sn (n ≥ 4). When n ≥ 4, we prove that the norm of its Higgs field is 1 and the connection is actually Yang–Mills. More precisely, its curvature vanishes when n ≥ 5. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang–Mills–Higgs pairs over a four-dimensional compact manifold with uniformly bounded energy. We show there is a subsequence that converges smoothly to a Yang–Mills–Higgs pair up to gauge modulo finitely many four-dimensional spheres with Yang–Mills connections.

Invariant Volume Forms of Geodesic, Potential, and Dissipative Systems on a Tangent Bundle of a Four-Dimensional Manifold

Invariant Volume Forms of Geodesic, Potential, and Dissipative Systems on a Tangent Bundle of a Four-Dimensional Manifold

Critical metrics for quadratic curvature functionals on some solvmanifolds

We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions G_{mathcal S}=H rtimes exp ({mathbb {R}}S) of the Heisenberg Lie group H, for any mathcal S in {mathfrak {s}}{mathfrak {p}}(1,mathbb R), equipped with a family g_a of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when g_a is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.

INVARIANT VOLUME FORMS OF GEODESIC, POTENTIAL, AND DISSIPATIVE SYSTEMS ON A TANGENT BUNDLE OF A FOUR-DIMENSIONAL MANIFOLD

Complete set of invariant differential forms of phase volume for homogeneous dynamical systems on tangent bundles to smooth four-dimensional manifolds are presented in this paper. The connection between the presence of these invariants and the complete set of the first integrals which are necessary for the integration of geodesic, potential and dissipative systems is shown. At the same time, the introduced force fields make the considered systems dissipative with dissipation of different signs and generalize the previously considered ones.