Weyl Geometry Research Articles

NON-RIEMANNIAN GEOMETRIES AND HIGHER-DIMENSIONAL THEORIES OF SPACETIME

We discuss the role that non-Riemannian geometries might play in the formulation of modern higher-dimensional spacetime theories. Among the vast number of non-Riemannian geometries we consider the two simplest ones: Weyl geometry and Riemann-Cartan geometry. We approach the problem of classical confinement of particles in branes and show how this question may be investigated in the context of the two mentioned geometries. We show that there is a classical analogue of the confinement of fermions induced by a scalar field and the one that might be induced by a Weyl field. In similar fashion we show that the same role may be played by a torsion field.

The cosmological constant and Pioneer anomaly from Weyl spacetimes and Mach's principle

It is shown how Weyl's geometry and Mach's holographic principle furnishes both the magnitude and sign (towards the sun) of the Pioneer anomalous acceleration aP∼−c2/RHubble firstly observed by Anderson et al. Weyl's geometry can account for both the origins and the value of the observed vacuum energy density (dark energy). The source of dark energy is just the dilaton-like Jordan–Brans–Dicke scalar field that is required to implement Weyl invariance of the most simple of all possible actions. A nonvanishing value of the vacuum energy density of the order of 10−123MPlanck4 is found consistent with observations. Weyl's geometry accounts also for the phantom scalar field in modern Cosmology in a very natural fashion.

On the embedding of space-time in five-dimensional Weyl spaces

We revisit Weyl geometry in the context of recent higher-dimensional theories of space-time. After introducing the Weyl theory in a modern geometrical language we present some results that represent extensions of Riemannian theorems. We consider the theory of local embeddings and submanifolds in the context of Weyl geometries and show how a Riemannian space-time may be locally and isometrically embedded in a Weyl bulk. We discuss the problem of classical confinement and the stability of motion of particles and photons in the neighborhood of branes for the case when the Weyl bulk has the geometry of a warped product space. We show how the confinement and stability properties of geodesics near the brane may be affected by the Weyl field. We construct a classical analog of quantum confinement inspired in theoretical-field models by considering a Weyl scalar field which depends only on the extra coordinate.

HKT geometry and de Sitter supergravity

Solutions of five-dimensional minimal de Sitter supergravity admitting Killing spinors are considered. It is shown that the “timelike” solutions are determined in terms of a four-dimensional hyper-Kähler torsion (HKT) manifold. If the HKT manifold is conformally hyper-Kähler the most general solution can be obtained from a sub-class of supersymmetric solutions of minimal N = 2 ungauged supergravity, by means of a simple transformation. Examples include a multi-BMPV de Sitter solution, describing multiple rotating black holes co-moving with the expansion of the universe. If the HKT manifold is not conformally hyper-Kähler, examples admitting a tri-holomorphic Killing vector field are constructed in terms of certain solutions of three-dimensional Einstein–Weyl geometry.

Localization of matters on pure geometrical thick branes

In the literatures, several types of thick smooth brane configurations in a pure geometric Weyl integrable 5-dimensional space time have been presented. The Weyl geometry is a non-Riemannian modification of 5-dimensional Kaluza-Klein (KK) theory. All these thick brane solutions preserve 4-dimensional Poincaré invariance, and some of them break Z2-symmetry along the extra dimension. In this paper, we study localization of various matter fields on these pure geometrical thick branes, which also localize the graviton. We present the shape of the potential of the corresponding Schrödinger problem and obtain the lowest KK mode. It is shown that, for both spin 0 scalars and spin 1 vectors, there exists a continuum gapless spectrum of KK states with m2 > 0. But only the massless mode of scalars is found to be normalizable on the brane. However, for the massless left or right chiral fermion localization, there must be some kind of Yukawa coupling. For a special coupling, there exist a series of discrete massive KK modes with m2 > 0. It is also showed that for a given coupling constant only one of the massless chiral modes is localized on the branes.

IMPORTANT REMARKS ON (ANTI) de SITTER SPACES AND THE COSMOLOGICAL CONSTANT

A class of proper and novel generalizations of the (anti) de Sitter solutions (parametrized by a family of radial functions R(r)) are presented that could provide a very plausible resolution of the cosmological constant problem along with a natural explanation of the ultraviolet/infrared (uv/ir) entanglement required to solve this problem. A nonvanishing value of the vacuum energy density of the order of [Formula: see text] is derived in agreement with the experimental observations. The presence of the radial function R(r) is instrumental to understand why the cosmological constant is not zero and why it is so tiny. The correct lower estimate of the mass of the observable universe related to the Dirac–Eddington's large number N = 1080is also obtained. Finally we present our most recent findings of how Weyl Geometry via a Brans–Dicke scalar field solves the riddle of dark energy in addition to providing another derivation of the vacuum energy density.

CR Weyl geometry and connections of the canonical bundle

A nondegenerate CR manifold has a natural conformal class consisting of hermitian forms, which are called Levi forms, on the hyperdistribution. It is important to study the objects on CR manifolds which are independent of the choice of Levi forms. A CR Einstein-Weyl structure defined in [8] is one of such objects. Our purpose is to find nondegenerate CR manifolds with CR Einstein-Weyl structures. In this paper, we shall obtain many examples of CR Einstein-Weyl manifolds. These examples are actually derived from the fact that CR Einstein-Weyl structures are closely concerned with connections of the canonical bundle of CR manifolds.

The twistor theory of Whitham hierarchy

We have generalized the approach of M. Dunajski, L. Mason and P. Tod [Einstein–Weyl geometry, the dKP equation and twistor theory, J. Geom. Phys. 37 (2001) 63–93] and established a 1–1 correspondence between a solution of the Universal Whitham hierarchy [I.M. Krichever, The τ -function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47 (1994) 437–475] and a twistor space. The twistor space consists of a complex surface and a family of complex curves together with a meromorphic 2-form. The solution of the Universal Whitham hierarchy is given by deforming the curve in the surface. By treating the family of algebraic curves in C P 1 × C P 1 as a twistor space, we were able to express the deformations of the isomonodromic spectral curve in terms of the deformations generated by the Universal Whitham hierarchy.

WEYL GEOMETRY QUANTIZATION IN SOLAR SYSTEM

In the present work we show that planetary mean distances can be calculated through considering the Weyl geometry. We interpret the Weyl gauge field as a vector field associated with the hypercharge of the particles and apply the gauge concept of the Weyl geometry. The results obtained are shown to agree with the observed orbits of all the planets and of the asteroid belt in the solar system, with some empty states.

Can Weyl geometry explain acceleration?

We study homogeneous and isotropic cosmologies in a Weyl spacetime. We show that for homogeneous and isotropic spacetimes, the field equations can be reduced to the Einstein equations with a two-fluid source. We write the equations as a two-dimensional dynamical system and analyze the qualitative, asymptotic behavior of the models. We examine the possibility that in certain theories the Weyl 1-form may give rise to a late accelerated expansion of the Universe and conclude that such behaviour is not met as a generic feature of the simplest cosmologies.

Differential equations and conformal structures

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of third order ODEs considered modulo contact transformations of variables and (local) three-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of third order ODEs satisfying the Wuenschmann and the Cartan conditions define a three-dimensional Lorentzian–Einstein–Weyl geometry. The third example associates to each point equivalence class of third order ODEs a six-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of second order ODEs and four-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature (3, 2). The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group G 2 . All the examples are deeply rooted in Elie Cartan’s works on exterior differential systems.

Gravitation and Electrodynamics over SO(3,3)

In a series of papers, an approach to field theory is developed in which matter appears by interpreting source-free (homogeneous) fields over a 6-dimensional space of signature (3,3), as interacting (inhomogeneous) fields in space-time. The extra dimensions are given a physical meaning as “coordinatized matter.” The inhomogeneous energy-momentum relations for the interacting fields in space-time are automatically generated by the simple homogeneous relations in 6-d. We then develop a Weyl geometry over SO(3,3) as base, under which gravity and electromagnetism are essentially unified via an irreducible 6-calibration invariant Lagrange density and corresponding variational principle. The Einstein–Maxwell equations are shown to represent a low-order approximation, and the cosmological constant must vanish in order that this limit exist.

Weyl structures with positive Ricci tensor

We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian–Weyl manifolds with non-negative symmetric part of the Ricci tensor of the canonical Weyl connection and show that every such manifold has first Betti number b 1=1 and Hodge numbers h p,0 =0 for p>0, h 0,1=1, h 0, q =0 for q>1.

Local Scale Invariance and General Relativity

According to the theory of unimodular relativity developed by Anderson and Finkelstein, the equations of general relativity with a cosmological constant are composed of two independent equations, one which determines the null-cone structure of space-time, another which determines the measure structure of space-time. The field equations that follow from the restricted variational principle of this version of general relativity only determine the null-cone structure and are globally scale-invariant and scale-free. We show that the electromagnetic field may be viewed as a compensating gauge field that guarantees local scale invariance of these field equations. In this way, Weyl's geometry is revived. However, the two principle objections to Weyl's theory do not apply to the present formulation: the Lagrangian remains first order in the curvature scalar and the nonintegrability of length only applies to the null-cone structure.

Yang–Mills theory and conjugate connections

We develop a new Yang–Mills theory for connections D in a vector bundle E with bundle metric h, over a Riemannian manifold by dropping the customary assumption Dh=0. We apply this theory to Einstein–Weyl geometry (cf. M.F. Atiyah, et al., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London 362 (1978) 425–461, and H. Pedersen, et al., Einstein–Weyl deformations and submanifolds, Internat. J. Math. 7 (1996) 705–719) and to affine differential geometry (cf. F. Dillen, et al., Conjugate connections and Radon's theorem in affine differential geometry, Monatshefts für Mathematik 109 (1990) 221–235). We show that a Weyl structure ( D, g) on a 4-dimensional manifold is a minimizer of the functional (D,g)↦ 1 2 ∫ M‖R D‖ 2v g if and only if ∗R D=±R D ∗ , where D ∗ is conjugate to D. Moreover, we show that the induced connection on an affine hypersphere M is a Yang–Mills connection if and only if M is a quadratic affine hypersurface.

Einstein–Weyl spaces and dispersionless Kadomtsev–Petviashvili equation from Painlevé I and II

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlev\'e transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are constant on a central quadric. We show how the solutions to the dKP equations can be used to construct some three-dimensional Einstein--Weyl structures, and four--dimensional anti-self-dual null-K\"ahler metrics.

Gauge Theory, Isotropy, and Surfaces in Affine 4-Space

The surface theory in the equiaffine space R4 is developed on the basis of H. Weyl’s gauge theory. Rescaling of the Weyl geometry leads to a 1-parameter family of invariant transversal plane bundles containig former special constructions. A transversal bundle metric is gained via the notion of isotropy. The paper then proceeds with a general tensorial theory, including theorema egregium and Radon-type results and a discussion of cubic fundamental forms. Finally there is given an application to homogeneous surfaces.

Almost contact Einstein-Weyl structures

Almost contact Weyl manifolds are introduced: in dimension at least 5 they naturally lead to locally conformal cosymplectic spaces. We analyze them from the point of view of Weyl geometry considering in particular the case of compact Einstein–Weyl manifolds.

Geometry of quaternionic Kähler connections with torsion

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess–Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in SP( n)·SP(1), QKT connection. We study the geometry of QKT connections. We find conditions to the existence of a QKT connection and prove that if it exists it is unique. We show that QKT geometry persist in a conformal class of metrics which allows us to obtain a lot of (compact) examples of QKT manifolds. We present a (local) description of four-dimensional homogeneous QKT structures relying on the known result of naturally reductive homogeneous Riemannian manifolds. We consider Einstein-like QKT manifold and find closed relations with Einstein–Weyl geometry in dimension 4.

Applications of Riemannian and Einstein–Weyl Geometry in the Theory of Second-Order Ordinary Differential Equations

We consider some properties of four-dimensional Riemannian spaces whose metric coefficients are associated with the coefficients of second-order nonlinear differential equations, and we study the properties of three-dimensional Einstein–Weyl spaces related to the dual equations b″ = g(a,b,b′), where the function g(a,b,b′) satisfies a special partial differential equation.