Positivity of energy inn-dimensional Lorentzian spaces

The spinor approach of Witten and Nester is used to show that in ann-dimensional Lorentzian spaceV _n , the conserved total energy-momentum vectorP is nonspace-like. It is shown thatP may be expressed in terms of an integral of an antisymmetric tensorE over ann – 2-dimensional subspace at space-like infinity inV _n .E is expressed in terms of a spinor field inV _n and its covariant derivatives. This tensor is a generalization of that used in the discussion of five-dimensional Lorentzian spaces.     

Symmetries of coset spaces and Kaluza-Klein supergravity

Known theorems about the isometry group of a general coset space $\text{G}\text{H}$ are reviewed. The Killing vectors on $\text{G}\text{H}$ are explicitly constructed. Rescalings of the coset vielbeins are discussed, and a simple criterion to find which rescalings preserve the isometry group is given. A general expression for the Riemann and Ricci tensors in terms of the rescaled vielbeins and the group structure constants is derived. These results have useful applications in Kaluza-Klein theories. As an example, the round and the squashed seven-spheres that have been used to compactify d = 11 supergravity are discussed, and it is shown that they can be identified with two appropriately rescaled coset spaces $\text{SO(5)}\text{SO(3)}$.     

Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces \({\mathbb {R}}^{N_1} \times _{\mathcal {R}} {\mathbb {R}}^{N_2}\). These coordinate algebras are quadratic ones associated with an \(\mathcal {R}\)-matrix which is involutive and satisfies the Yang–Baxter equations. As a consequence, they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces \({\mathbb {R}}^{4} \times _{\mathcal {R}} {\mathbb {R}}^{4}\). Among these, particularly well behaved ones have deformation parameter \(\mathbf{u} \in {\mathbb {S}}^2\). Quotients include seven spheres \({\mathbb {S}}^{7}_\mathbf{u}\) as well as noncommutative quaternionic tori \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u} = {\mathbb {S}}^3 \times _\mathbf{u} {\mathbb {S}}^3\). There is invariance for an action of \({{\mathrm{SU}}}(2) \times {{\mathrm{SU}}}(2)\) on the torus \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u}\) in parallel with the action of \(\mathrm{U}(1) \times \mathrm{U}(1)\) on a ‘complex’ noncommutative torus \({\mathbb {T}}^2_\theta \) which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.     

On the spectrum of Kaluza-Klein theories with non-compact internal spaces

We investigate the conditions under which Kaluza-Klein theories with non-compact internal spaces admit a discrete spectrum.     

Biminimal properly immersed submanifolds in the Euclidean spaces

We consider a complete nonnegative biminimal   submanifold M$M$ (that is, a complete biminimal submanifold with λ≥0$\lambda \ge 0$) in a Euclidean space E^N${\mathbb{E}}^N$. Assume that the immersion is proper  , that is, the preimage of every compact set in E^N${\mathbb{E}}^N$ is also compact in M$M$. Then, we prove that M$M$ is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0$\lambda <0$, we construct examples of biminimal submanifolds and curves.     

Algebraic classification of four-dimensional locally Euclidean spaces

The algebraic classification of locally Euclidean four-dimensional spaces (having positive-definite metric) is considered in terms of dual complex variables, which, like ordinary complex numbers, are a type of hypercomplex numbers. It is shown that only four space subtypes exist: 1, 1a, D, 0 and there are no wave types. Transition between space subtypes is described by a CUSP catastrophe analogous to a second-order phase transition in a solid. Introducing an imaginary coordinate, we obtain the usual four-dimensional space—time, and our classification reduces to the well-known Petrov—Penrose algebraic classification of gravitational fields.The work was carried out under the auspices of the Interindustry Science and Technology Astronomy Program.Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, 90–95, November, 1994.     

The evolution of the universe from non-compact Kaluza-Klein theory

We develop a 5D mechanism, inspired by Campbells theorem, to explain the (neutral scalar field governed) evolution of the universe from an initially inflationary expansion that has a change of phase towards a decelerated expansion and thereafter evolves towards the present day observed accelerated (quintessential) expansion.Received: 4 October 2004, Published online: 23 November 2004PACS: 04.20.Jb, 11.10.Kk, 98.80.Cq     

Freeq-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group

After a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups, the quantum hyperboloidqH and the quantum planeqP are determined as homogeneous spaces ofF _q (E(2)). The canonical action ofE _q (2) is used to define a naturalq-analog of the free Schrödinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of twoq-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in theqP case, are given in terms of Hahn-Exton functions. Introducing the universalT-matrix forE _q (2) we prove that the Hahn-Exton as well as Jacksonq-Bessel functions are also obtained as matrix elements ofT, thus giving the correct extension to quantum groups of well known methods in harmonic analysis.     

Metrics on the moduli spaces of instantons over Euclidean 4-space

We prove that the natural hyper-Kähler metrics on the moduli space of chargek instantons over Euclidean four-space and on the space of ADHM matrices coincide. We use this to deduce formulae relating expressions in the curvature of a connection to invariant polynomials in the ADHM matrices corresponding to this connection. These arise from consideration of the group of symmetries acting on the moduli spaces.     

On the existence of hydrogen atoms in higher dimensional Euclidean spaces

The question of whether hydrogen atoms can exist or not in spaces with a number of dimensions (D) greater than 3 is revisited. The lowest quantum mechanical stable states and the corresponding wave functions are determined by applying Numerov?s method to solve Schrödinger?s equation. States for different angular momentum quantum number and dimensionality are considered. One is lead to the result that hydrogen atoms in higher dimensions could actually exist. The most probable distance between the electron and the nucleus are then computed as a function of D showing the possibility of tiny confined states.     

On the theory of algebraic invariants of vector spaces of Killing tensors

We employ a hybrid version of the classical Cartan method of moving frames to develop a practical algorithm for the generation of isometry group invariants and covariants for arbitrary vector spaces of Killing tensors defined on any flat pseudo-Riemannian manifold. We then apply our algorithm to construct a set of fundamental covariants for the space of valence-two Killing tensors defined in three-dimensional Euclidean space and use them to invariantly characterize the associated eleven orthogonally separable webs.     

The geodesics of the Kaluza-Klein wormhole soliton

The Kaluza-Klein wormhole soliton metric is a regular localized solution with Minkowskian signature, to the sourceless five-dimensional Einstein equations. We apply five-to-three dimensional reduction to convert the problem of geodesic motion of neutral or charged test particles in this metric to a non-relativistic potential problem, which we discuss in detail, studying bound and scattering states. We show that there is no observable difference between scattering of a spinless test particle by a point charge and by a wormhole soliton.     

On the explicit construction of Parisi landscapes in finite dimensional Euclidean spaces

AnN-dimensional Gaussian landscape with multiscale translation-invariant logarithmic correlations has been constructed, and the statistical mechanics of a single particle in this environment has been investigated. In the limit of a high dimensional N → ∞, the free energy of the system in the thermodynamic limit coincides with the most general version of Derrida’s generalized random energy model. The low-temperature behavior depends essentially on the spectrum of length scales involved in the construction of the landscape. The construction is argued to be valid in any finite spatial dimensions N ≥1. The text was submitted by the authors in English.     

The motion of Kaluza-Klein monopoles

A scheme is proposed and justified for examining the motion of the five dimensional Kaluza-Klein monopoles at low energy. The classical and quantum scattering is discussed and it is shown that for all separations and at small velocities the monopoles do not interact with one another.     

The upperbound of the volume expansion rate in a Lorentzian manifold

Using the differential equation obtained from spacelike level hypersurfaces in a Lorentzian manifold, the volume expansion rate of an achronal spacelike hypersurface orthogonal to a timelike geodesic is investigated in terms of the integral Ricci and scalar curvature bound.     

Chern-Simons-Witten invariants of lens spaces and torus bundles,and the semiclassical approximation

We derive explicit formulas for the Chern-Simons-Witten invariants of lens spaces and torus bundles overS ¹, for arbitrary values of the levelk. Most of our results are for the groupG=SU(2), though some are for more general compact groups. We explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the Chern-Simons path integral.Partially supported by an NSF Graduate FellowshipAddress as of September 1, 1991: School of Natural Science, Institute for Advanced Study, Princeton, NJ 08540; USA     

Modification of the Coulomb Potential from a Kaluza-Klein Model with a Gauss-Bonnet Term in the Action

In four dimensions a Gauss-Bonnet term in the action corresponds to a total derivative, and therefore it does not contribute to the classical equations of motion. For higher-dimensional geometries this term has the interesting property (which it shares with other dimensionally continued Euler densities) that when the action is varied with respect to the metric, it gives rise to a symmetric, covariantly conserved tenser of rank two which is a function of the metric and its first- and second-order derivatives. Here we review the unification of general relativity and electromagnetism in the classical five-dimensional, restricted (with g₅₅ = 1) Kaluza-Klein model. Then we discuss the modifications of the Einstein-Maxwell theory that results from adding the Gauss-Bonnet term in the action. The resulting four-dimensional theory describes a non-linear U(1) gauge theory non-minimally coupled to gravity. For a point charge at rest we find a perturbative solution for large distances which gives a mass-dependent correction to the Coulomb potential. Near the source we find a power-law solution which seems to cure the short-distance divergency of the Coulomb potential. Possible ways to obtain an experimental upper limit to the coupling of the hypothetical Gauss-Bonnet term are also considered.     

Kaluza-Klein theory from the viewpoint of gauge theory of gravity

We consider Kaluza-Klein theory based on the fiber bundle. We obtain the modified Kaluza-Klein metric as an invariant line element of a bundle. Its reduced action includes a higher derivative action in gravitation as well as a term linear inR.     

Estimates for the Volume of a Lorentzian Manifold

We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume.