non-Riemannian Spaces Research Articles

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId)−1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators Cc∞(X)→D′(X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D′(X) forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R(z) extends, and the residue representations can be infinite-dimensional.

The geometry, branes and applications of exceptional field theory

This is a review of exceptional field theory: a generalisation of Kaluza–Klein theory that unifies the metric and [Formula: see text]-form gauge field degrees of freedom of supergravity into a generalised or extended geometry, whose additional coordinates may be viewed as conjugate to brane winding modes. This unifies the maximal supergravities, treating their previously hidden exceptional Lie symmetries as a fundamental geometric symmetry. Duality orbits of solutions simplify into single objects, that in many cases have simple geometric interpretations, for instance as wave or monopole-type solutions. It also provides a route to explore exotic or nongeometric aspects of M-theory, such as exotic branes, [Formula: see text]-folds, and more novel sorts of non-Riemannian spaces.

Photon mass limits driven by axions, Lorentz violation, chiral and gravitational anomalies in non-Riemannian spaces, and helical dynamo seeds

In this paper two types of applications of Cartan torsion to cosmology and quantum field theory are given. The first is given by massive photon axial torsion mixing through non-minimal coupling. It is shown that chiral anomalies induce upper limits on the photon mass and galactic dynamo seeds. In this example we obtain magnetic field seeds of the order of 10−16 G from a strong chiral battery magnetic field of the order of 1030 G. This cosmic magnetic field is in accordance with previous magnetic dynamo seed limits as 10−22 G ≤ Bseed ≤ 10−12 G given by Barrow et al. (Phys. Rev. D, 86, 023533 (2012). doi: 10.1103/PhysRevD.86.023533 ). In this example, as in the Prokopec et al. (Phys. Rev. Lett. 89, 101301 (2002). doi: 10.1103/PhysRevLett.89.101301 ) torsionless case, which associates Dolgov anomalies to photon mass limits, we show that at early universe scales, 10−32 eV is a very stringent limit. Here we can show that the parameter [Formula: see text], which is well within the lower bound of r needed for the magnetic field to seed galactic dynamos. Helical magnetic fields are obtained. One can also derive massive photon limits from axial chiral anomalies with torsion degrees of freedom (Duncan et al. Nucl. Phys. B, 387, 215 (1992). doi: 10.1016/0550-3213(92)90052-D ) leading to photon mass of the order of 10−11 eV. Dynamo equations are obtained from anomalous Maxwell equations as well as magnetic wave equations with photon mass, which are analytically solved. In the second example one shows that Chern–Simons (CS) current is affected by the presence of minimal coupled torsion. In this case as well one obtains a photon mass limit of the order 10−18 eV. Gravitational anomalies in Riemann–Cartan spacetime with axial torsion are also obtained. Recently, Terrano et al. (Phys. Rev. Lett. 122, 231301 (2019). doi: 10.1103/PhysRevLett.122.231301 ) computed dark matter mass constraints of the order of 10−23 eV from spin-polarised torsion balances, which has given some support for torsion theories of Einstein–Cartan type. Here, however, we deal with torsion with dynamical degrees of freedom. Lorentz violation in non-Riemannian CS currents are investigated.

Minimal coupling in presence of non-metricity and torsion

We deal with the question of what it means to define a minimal coupling prescription in presence of torsion and/or non-metricity, carefully explaining while the naive substitution partial rightarrow nabla introduces extra couplings between the matter fields and the connection that can be regarded as non-minimal in presence of torsion and/or non-metricity. We will also investigate whether minimal coupling prescriptions at the level of the action (MCPL) or at the level of field equations (MCPF) lead to different dynamics. To that end, we will first write the Euler–Lagrange equations for matter fields in terms of the covariant derivatives of a general non-Riemannian space, and derivate the form of the associated Noether currents and charges. Then we will see that if the minimal coupling prescriptions is applied as we discuss, for spin 0 and 1 fields the results of MCPL and MCPF are equivalent, while for spin 1/2 fields there is a difference if one applies the MCPF or the MCPL, since the former leads to charge violation.

Kaluza–Klein reduction on a maximally non-Riemannian space is moduli-free

We propose a novel Kaluza–Klein scheme which assumes the internal space to be maximally non-Riemannian, meaning that no Riemannian metric can be defined for any subspace. Its description is only possible through Double Field Theory but not within supergravity. We spell out the corresponding Scherk–Schwarz twistable Kaluza–Klein ansatz, and point out that the internal space prevents rigidly any graviscalar moduli. Plugging the same ansatz into higher-dimensional pure Double Field Theory and also to a known doubled-yet-gauged string action, we recover heterotic supergravity as well as heterotic worldsheet action. In this way, we show that 1) supergravity and Yang–Mills theory can be unified into higher-dimensional pure Double Field Theory, free of moduli, and 2) heterotic string theory may have a higher-dimensional non-Riemannian origin.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

Homogeneous space with non-virtually abelian discontinuous groups but without any proper SL(2, ℝ)-action

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of “continuous analogue” gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space [Formula: see text] admits a discontinuous group which is not virtually abelian if and only if [Formula: see text] admits a proper [Formula: see text]-action (T. Okuda [J. Differ. Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces [Formula: see text] of reductive type. We give a counterexample in the case [Formula: see text].

Estimate of observable diameter of l p -product spaces

We estimate the observable diameter of the lp-product space Xn of an mm-space X by using the limit formula in our previous paper (Ozawa and Shioya, Math. Zeit. (to appear)). The idea of our proof is based on Gromov’s book (In: Metric structures for Riemannian and non-Riemannian spaces, 2007). As a corollary we obtain the phase transition property of \({\{X^n\}_{n=1}^\infty}\) under a uniform discreteness condition.

Ricci Flow on Modified Riemann Extensions

The Ricci flow and the evolution equations of the Riemannian curvature tensor were initially introduced by Hamilton [8] and was later studied to a large extent by Perelman [13–15], Cao and Zhu [4], Morgan and Tian [10]. Indeed, the theory of Ricci flow has been used to prove the geometrization and Poincare conjectures [1]. However not much work has been done on Ricci flows on modified Riemann extensions. The Ricci flow equation is the evolution equation ∂gij ∂t = −2Rij where gij and Rij are metric components respectively. As flow progresses the metric changes and hence the properties related to it. Patterson and Walker [11] have defined Riemann extensions and showed how a Riemannian structure can be given to the 2n dimensional tangent bundle of an n-dimensional manifold with given non-Riemannian structure. This shows that Riemann extension provides a solution of the general problem of embedding a manifold M carrying a given structure in a manifold M carrying another structure, the embedding being carried out in such a way that the structure on M induces in a natural way the given structure on M . The Riemann extension of Riemannian or non-Riemannian spaces can be constructed with the help of the Christoffel coefficients Γijk of corresponding Riemann space or with connection coefficients Π i jk in the case of the space of affine connection [5]. The theory of Riemann extensions has been extensively studied by Afifi [1]. Though the Riemann extensions itself is rich in geometry, here in our discussions, the modified Riemann extensions fit naturally in to the frame work. Modified Riemann extensions are introduced in [2] and their properties we list briefly in the next section.

SYMMETRIC TELEPARALLEL GRAVITY: SOME EXACT SOLUTIONS AND SPINOR COUPLINGS

In this paper, we elaborate on the symmetric teleparallel gravity (STPG) written in a non-Riemannian space–time with nonzero nonmetricity, but zero torsion and zero curvature. First, we give a prescription for obtaining the nonmetricity from the metric in a peculiar gauge. Then, we state that under a novel prescription of parallel transportation of a tangent vector in this non-Riemannian geometry, the autoparallel curves coincide with those of the Riemannian space–times. Subsequently, we represent the symmetric teleparallel theory of gravity by the most general quadratic and parity conserving Lagrangian with lagrange multipliers for vanishing torsion and curvature. We show that our Lagrangian is equivalent to the Einstein–Hilbert Lagrangian for certain values of coupling coefficients. Thus, we arrive at calculating the field equations via independent variations. Then, we obtain in turn conformal, spherically symmetric static, cosmological and pp-wave solutions exactly. Finally, we discuss a minimal coupling of a spin-1/2 field to STPG.

Cross-Modal Perception in the Framework of Non-Riemannian Sensory Space

Though human sensations, such as the senses of hearing, sight, etc., are independent each other, the interference between two of them is sometimes observed, and is called cross-modal perception[1]....

Probability distribution of metric measure spaces

In this paper we are going to generalize Gromov's mm-Reconstruction theorem (cf. [Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, Basel, 1999] 3 1 2 . 5 ) to a probability measures on the spaces of mm-spaces. And for this purpose, we give alternative proof of mm-Reconstruction theorem.

Book Review: Metric structures for Riemannian and non-Riemannian spaces

Book Review: Metric structures for Riemannian and non-Riemannian spaces

Spacetime defects: Domain walls and torsion

The theory of distributions in non-Riemannian spaces is used to obtain exact static thin domain wall solutions of Einstein-Cartan equations of gravity. Curvature δ-singularities are found while Cartan torsion is given by Heaviside functions. Weitzenböck planar walls are characterized by torsion δ-singularities and zero curvature. It is shown that Weitzenböck static thin domain walls do not exist exactly as in general relativity. The global structure of Weitzenböck nonstatic torsion walls is investigated.

Novel Results in the Geometry of Geodesics

This work is devoted to results on the G-spaces of Busemann found after the appearance of his book "The Geometry of Geodesics" (Academic Press, New York, 1955); and denoted here as G. The axioms for G-spaces are found on p. 37 of G. They include all complete Riemann spaces and many others which the letter G denotes as Finsler spaces, which is a gross misnomer. Finsler was the first who investigated non-Riemannian spaces (under strong differentiability hypotheses) guided by Carathéodory, whose methods in the calculus of variations form the basis of Finsler′s thesis, which has no relation to our G-spaces (but is still an active field of research). The misnomer was caused by the fact that Finsler′s thesis (1918) was inaccessible until 1951, when Birkhäuser reissued it unchanged. Nevertheless, we denote here as Finsler spaces all non-Riemannian G-spaces, because we refer frequently to G and this greatly facilitates references.

Distorted Riemannian space and technique of differential geometry

A differential-geometric technique of non-Riemannian space description is considered. This n-dimensional distorted Riemannian space distinguishes from a Riemannian one by a substitution of one-dimensional geodesics by (n−1)-dimensional tubes. Such a space is of interest as a model of space-time, taking into account quantum effects.

Non-Riemannian approach to geometry of visual space: An application of affinely connected geometry to visual alleys and horopter

A study was undertaken to give a concrete model to the theory of non-Riemannian visual space developed with reference to Luneburg's Riemannian model, and to apply the model to visual alleys and horopter. The parallel and distance alleys were formulated to be geodesics in the non-Riemannian visual space under the restriction of teleparallelism and Riemannian configuration, respectively. The present model was described in terms of an affinely connected space which was one of the non-Riemannian spaces without the concept of length. On the supposition that the parallel alley and horopter are paths in visual space, an asymmetric connection was obtained under the restriction of teleparallelism. There are two options for carrying out the numerical simulation. In Option 1, to calculate the connection, families of curves were fitted to the data of parallel alley and horopter. By the numerical solutions of the equation of geodesics with Christoffel symbols which were deduced from the asymmetric connection, the distance alley was drawn independently of the experimental data. The resulting curve was found to lie outside the parallel alley as did Blumenfeld's experiment. One of the possibilities causing the discrepancy between parallel and distance alleys was tested in 1.3. Furthermore, two alleys are expected to coincide if points are adjusted in position, while the alleys are being constructed, along the horopter curves rather than the front-parallel lines. It was shown that the numerical solutions of the geodesic with the connection thus obtained lie midway between parallel and distance alleys. In Option 2, the metric tensor was obtained in the explicit form, and it became clear that the Gaussian curvature with respect to the metric is not constant.

Geometry of dislocated de Broglie waves

The geometrical structures implicit in the de Broglie waves associated with a relativistic charged scalar quantum mechanical particle in an external field are analyzed by employing the ray concept of the causal interpretation. It is shown how an osculating Finslerian metric tensor, a torsion tensor, and a tetrad field define respectively the strain, the dislocation density, and the Burgers vector in the “natural state” of the wave, which is a non-Riemannian space of distant parallelism. A quantum torque determined by the quantum potential is introduced and the example of a screw dislocated wave is discussed.

Differential geometry of the coincidence site lattice

The tensor properties of simple internal surfaces, such as two-phase interfaces and grain boundaries, have been studied in detail. In particular, these tensor quantities have been defined with respect to the original crystal lattice as well as to a common coincidence site lattice that is a characteristic of the boundary. Such lattices allow a given type of distortion to be represented in either a Riemannian or a non-Riemannian (dislocated) space. This particular generality provides a powerful method of anlysing problems in continuum mechanics.

Electromagnetism in non-riemannian space

Electromagnetism in non-riemannian space