Four-dimensional Submanifold Research Articles

A note on the Kaluza–Klein theory

We show that the Kaluza–Klein (KK) theory contains a fundamental problem: The four-dimensional metric tensor and the electromagnetic potential vector assumed in the KK theory belong to four-dimensional vector spaces that are not integrable in general, resulting that the four-dimensional physical variables and the corresponding field equations derived from the five-dimensional Einstein field equation (i.e. the four-dimensional Einstein field equation and the Maxwell equations) are not defined on a four-dimensional submanifold. That is, the four-dimensional spacetime assumed in the KK theory does not exist. No satisfactory solutions are found within the KK formalism. Perhaps the best approach to fix the problem is giving up the KK theory and looking for a new unified scheme for gravitational and electromagnetic interactions in the framework of a spacetime with extra dimensions, as having already been explored in some literature.

Lagrangian-type submanifolds of G2 and Spin(7) manifolds and their deformations

Three-dimensional Harvey-Lawson submanifolds were introduced in an earlier paper by Akbulut and Salur [1], as examples of Lagrangian-type manifolds inside a G2 manifold. In this paper, we discuss these as well as two other similar types of submanifolds of G2 and Spin(7) manifolds and their deformations. We first show that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. We then introduce a new class of Lagrangian-type four-dimensional submanifolds of M, call them RS-submanifolds, and prove that the space of deformations of a smooth, compact, orientable RS-submanifold of a G2 manifold M can be identified with the space of closed 3-forms on RS. Finally, we describe an analogous setting for Spin(7) manifolds by defining a new class of Lagrangian-type four-dimensional submanifolds of a Spin(7) manifold N, which we call L-submanifolds. We show that the space of deformations of a smooth, compact, orientable L-submanifold of N can be identified with the space of closed 3-forms on L.

Line-symmetric motion generators

When a rigid body is axially reflected through a moving line, its image undergoes a so-called line-symmetricmotion. The space comprising all possible line-symmetric motions that share a common initial line is a four-dimensional submanifold, denoted M4, in the special Euclidean group SE(3). Recently, we showed that M4 may be used to characterize motions of a line-symmetric body that are free of self-spin and sliding, thus lending itself to applications such as remote center of motion devices for minimal invasive surgery and haptic interfaces. Aiming at designing robot mechanisms for these applications, we present in this paper a systematic enumeration of line-symmetric motion generators (LSMGs), i.e., robot mechanisms that generate the line-symmetric motion manifold M4, following a procedure based on symmetric space theory. LSMGs present a ubiquitous line symmetry of their joint axes, thus offering a new understanding of the line-symmetric motions.

New invariant relations for one critical subsystem of a generalized two-field gyrostat

In this paper new invariant relations for one critical subsystem of a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsyganov, known as a generalized two-field gyrostat, are obtained. The dynamic system that is induced on the invariant four-dimensional submanifolds is almost everywhere Hamiltonian with two degrees of freedom. The type of system motions on this invariant manifold is determined.

New invariant relations for the generalized two-field gyrostat

In the paper, we consider a completely integrable Hamiltonian system with three degrees of freedom found by V.V.Sokolov and A.V.Tsiganov. This system is known as the generalized two-field gyrostat. For the case of only gyroscopic forces present, we find new invariant four-dimensional submanifolds such that the induced dynamical systems are almost everywhere Hamiltonian with two degrees of freedom.

Phase topology of one irreducible integrable problem in the dynamics of a rigid body

We consider the integrable system with three degrees of freedom for which V. V. Sokolov and A. V. Tsiganov specified the Lax pair. The Lax representation generalizes the L-A pair found by A. G. Reyman and M. A. Semenov-Tian-Shansky for the Kovalevskaya gyrostat in a double field. We give explicit formulas for the additional first integrals K and G (independent almost everywhere), which are functionally related to the coefficients of the spectral curve for the Sokolov-Tsiganov L-A pair. Using this form of the additional integrals K and G and the Kharlamov parametric reduction, we analytically present two invariant four-dimensional submanifolds where the induced dynamical system is Hamiltonian (almost everywhere) with two degrees of freedom. The system of equations specifying one of the invariant submanifolds is a generalization of the invariant relations for the integrable Bogoyavlensky case (rotation of a magnetized rigid body in homogeneous gravitational and magnetic fields). We use the method of critical subsystems to describe the phase topology of the whole system. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of the Liouville tori both inside the subsystems and in the whole system.

Warped string compactification via singular Calabi–Yau conformal field theory

We construct spacetime supersymmetric, modular invariant partition functions of strings on the conifold-type singularities which include contributions from the discrete-series representations of SL(2,R). The discrete spectrum is automatically consistent with the GSO projection in the continuous sector, and contains massless matter fields localized on a four-dimensional submanifold at the tip of a cigar. In particular, they are in the 27⊕1 of E6 for the E8×E8 heterotic string. We speculate about a possible realization of local E6 GUT by using this framework.

Holomorphic classification of four-dimensional surfaces in

We use the method of model surfaces to study real four-dimensional submanifolds of . We prove that the dimension of the holomorphic symmetry group of any germ of an analytic four-dimensional manifold does not exceed 5 if this dimension is finite. (There are only two exceptional cases of infinite dimension.) The envelope of holomorphy of the model surface is calculated. We construct a normal form for arbitrary germs and use it to give a holomorphic classification of completely non-degenerate germs. It is shown that the existence of a completely non-degenerate CR-structure imposes strong restrictions on the topological structure of the manifold. In particular, the four-sphere admits no completely non-degenerate embedding into a three-dimensional complex manifold.

GRAVITATIONAL AND COSMOLOGICAL PROPERTIES OF A BRANE-UNIVERSE

The aim of this contribution is to provide a short introduction to recently investigated models in which our accessible universe is a four-dimensional submanifold, or brane, embedded in a higher dimensional spacetime and ordinary matter is trapped in the brane. I focus here on the gravitational and cosmological aspects of such models with a single extra-dimension.

Black diamonds at brane junctions

We discuss the properties of black holes in brane-world scenarios where our universe is viewed as a four-dimensional sub-manifold of some higher-dimensional spacetime. We consider in detail such a model where four-dimensional spacetime lies at the junction of several domain walls in a higher dimensional anti-de Sitter spacetime. In this model there may be any number p of infinitely large extra dimensions transverse to the brane-world. We present an exact solution describing a black p-brane which will induce on the brane-world the Schwarzschild solution. This exact solution is unstable to the Gregory-Laflamme instability, whereby long-wavelength perturbations cause the extended horizon to fragment. We therefore argue that at late times a non-rotating uncharged black hole in the brane-world is described by a deformed event horizon in p+4 dimensions which will induce, to good approximation, the Schwarzschild solution in the four-dimensional brane world. When p=2, this deformed horizon resembles a black diamond and more generally for p>2, a polyhedron.

An exotic class of Kaluza-Klein models

We discuss an exotic class of Kaluza-Klein models in which the internal space is neither compact nor even of finite volume. Rather than using the usual compact internal space we consider the case where particles are gravitationally trapped near a four-dimensional submanifold of the higher-dimensional spacetime. A specific model exhibiting this phenomenon is constructed in five dimensions.

The Lagrange rigid body motion

by T. RATIU and P. van MOERBEKEIn this paper, we discuss the three-dimensional rigid body motionabout a fixed point under the influence of gravity; the main emphasiswill be put on its symplectic structure, its constants of the motionand the origin of these from a group theoretical point of view. It canbe expressed as a Hamiltonian vector field on the 6-dimensional spaceSO (3) x so (3). The in variance of the problem under rotation aboutthe direction of gravity and about the axis of symmetry, leads toconservation of angular momentum with regard to the gravity axis.This invariant together with a trivial extra-invariant leads to a reduc-tion of this problem to a smooth manifold symplectically diffeomor-phic to a four-dimensional submanifold of so (3 )x so (3). TheHamiltonian vector field in this reduced manifold leads preciselyto the customary Euler-Poisson equations M=Mxft+rxxand r = r x Sl, where M, i2, F and denote respectively theangular momentum, angular velocity, the coordinates of the unitvector in the direction of gravity and the coordinates of the centerof mass, all expressed in body-coordinates.Another description of the situation comes from consideringgeneric coadjoint orbits of the Lie algebra of the semi-direct productgroup SO (3) x 50 (3); this is to say the four-dimensional manifoldconsidered above appears among these orbits; according to theKostant-Kirillov-Souriau method, these orbits have a natural sym-plectic structure, which turns out to coincide with the one before;the invariants characterizing these orbits are exactly the ones dis-cussed previously. This will be the object of section 1.