Cartan Manifold Research Articles

Homological aspects of topological gauge-gravity equivalence

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

Quantitative evaluation of Frank vector on the basis of continuous field theory of dislocation and differential geometry

This study conducts the modeling and numerical analysis of kink deformation on the basis of continuous field theory of dislocation and differential geometry. In particular, we aim to evaluate the existence and nature of disclination which is expected to be formed at the tip of kink band. Plastic deformation due to dislocation is obtained by numerically solving the Cartan first structure equation. This result yields a Riemann - Cartan manifold which equips Riemannian metric and Levi – Civita connection. By introducing the holonomy, i.e., the integration of curvature on a closed interval, we evaluate the magnitude of Frank vector at the kink tip. The result shows quantitative agreement with those expected from the classical dislocation theory.

Micropolar continua as projective space of Skyrmions

ABSTRACT Micropolar continua are shown to be the generalisation of nematic liquid crystals through perspectives of order parameters, topological and geometrical considerations. Micropolar continua and nematic liquid crystals are recognised as antipodals of and in projective geometry. We show that position-dependent rotational axial fields in kinematic micropolar continua can be considered as solutions of anisotropic Higgs fields, characterised by integers N. We emphasise that the identical integers N are topological invariants through homotopy classifications based on defects of order parameters and a finite energy requirement. Magnetic monopoles and Skyrmions are investigated based on the theories of defects of continua in Riemann–Cartan manifolds.

Chaos in Cartan foliations.

Chaotic foliations generalize Devaney's concept of chaos for dynamical systems. The property of a foliation to be chaotic is transversal, i.e, depends on the structure of the leaf space of the foliation. The transversal structure of a Cartan foliation is modeled on a Cartan manifold. The problem of investigating chaotic Cartan foliations is reduced to the corresponding problem for their holonomy pseudogroups of local automorphisms of transversal Cartan manifolds. For a Cartan foliation of a wide class, this problem is reduced to the corresponding problem for its global holonomy group, which is a countable discrete subgroup of the Lie automorphism group of an associated simply connected Cartan manifold. Several types of Cartan foliations that cannot be chaotic are indicated. Examples of chaotic Cartan foliations are constructed.

Demystification of nonrelativistic theories in curved background

We discuss a new formalism for constructing a nonrelativistic (NR) theory in curved background. Named as galilean gauge theory, it is based on gauging the global galilean symmetry. It provides a systematic algorithm for obtaining the covariant curved spacetime generalisation of any NR theory defined in flat spacetime. The resulting background is just the Newton–Cartan manifold. The example of NR free particle is explicitly demonstrated.

Energy condition respecting warp drives: the role of spin in Einstein–Cartan theory

In this paper we study the so called ‘warp drive’ spacetimes within the U4 Riemann–Cartan manifolds of Einstein–Cartan theory. Specifically, the role that spin may play with respect to energy condition violation is considered. It turns out that with the addition of spin, the torsion terms in Einstein–Cartan gravity do allow for energy condition respecting warp drives. Limits are derived which minimize the amount of spin required in order to have a weak/null-energy condition respecting system. This is done both for the traditional Alcubierre warp drive as well as for the modified warp drive of Van Den Broeck which minimizes the amount of matter required for the drive. The ship itself is in a region of effectively vacuum and hence the torsion, which in Einstein–Cartan theory is localized in matter, does not affect the geodesic nature of the ship’s trajectory. We also comment on the amount of spin and matter required in order for these conditions to hold.

Prescribing the mixed scalar curvature of a foliated Riemann–Cartan manifold

The mixed scalar curvature is one of the simplest curvature invariants of a foliated Riemannian manifold. We explore the problem of prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibred manifold, we reduce the problem to solution of a leafwise elliptic equation, which has three stable solutions -- only one of them corresponds to the case of a foliated Riemannian manifold.

Invariant affine connections on odd-dimensional spheres

A Riemann–Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann–Cartan space is introduced in a natural way. The paper is focused on the case of the odd-dimensional spheres $$\mathbb S^{2n+1}$$ endowed with their canonical Riemannian round metrics and viewed as homogeneous spaces of the special unitary groups. The classical Nomizu’s Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the spheres $$\mathbb S^{2n+1}$$ into homogeneous Riemann–Cartan spaces. The expressions of such connections as covariant derivatives are given by means of several invariant tensors: the ones of the usual Sasakian structure of the sphere; an invariant 3-differential form coming from a 3-Sasakian structure on $$\mathbb S^7$$ ; and the involved ones in the almost contact metric structure of $$\mathbb S^5$$ provided by its natural embedding into the nearly Kahler manifold $$\mathbb S^6$$ . Furthermore, the invariant connections sharing geodesics with the Levi-Civita one have also been completely described. Finally, $$\mathbb S^3$$ and $$\mathbb S^7$$ are characterized as the unique odd-dimensional spheres which admit nontrivial invariant connections satisfying an Einstein-type equation.

Automorphisms of Riemann–Cartan manifolds

It is proved that the maximal dimension of the Lie group of automorphisms of an n-dimensional Riemann–Cartan manifold (space) (Mn, g, \(\tilde \nabla \)) equals n(n − 1)/2+ 1 for n > 4 and, if the connection \(\tilde \nabla \) is semisymmetric, for n ≥ 2. If n = 3, then the maximal dimension of the group equals 6. Three-dimensional Riemann–Cartan spaces (M3, g, \(\tilde \nabla \)) with automorphism group of maximal dimension are studied: the torsion s and the curvature \(\tilde k\) are introduced, and it is proved that s and \(\tilde k\) are characteristic constants of the space and \(\tilde k\) = k − s2, where k is the sectional curvature of the Riemannian space (M3, g); a geometric interpretation of torsion is given. For Riemann–Cartan spaces with antisymmetric connection, the notion of torsion at a given point in a given three-dimensional direction is introduced.

Groups of Automorphisms of Riemann–Cartan Structures

In this paper, we calculate the maximal dimension of the Lie group of automorphisms of an n-dimensional Riemann–Cartan manifold.

Geometry of product complex Cartan manifolds

Abstract In this paper we consider the product of two complex Cartan manifolds, the outcome being a class of product complex Cartan spaces. Then, we study the relationships between the geometric objects of a product complex Cartan space and its components, (e.g. Chern-Cartan complex nonlinear connection, Cartan tensors). By means of these, we establish the necessary and sufficient conditions under which a product complex Cartan space is Landsberg-Cartan or Berwald-Cartan or it has some other properties.

Vanishing theorems for some classes of Riemann–Cartan manifolds

In this paper, a classification of Riemann–Cartan manifolds based on the orthogonal decomposition of the torsion tensor is given. Problems on the existence of two classes ℘1 ⨁ ℘2 and ℘3 of Riemann–Cartan spaces are discussed.

Riemann–Cartan manifolds

Riemann–Cartan manifolds

Pseudo-killing and pseudoharmonic vector fields on a Riemann—Cartan manifold

Six classes of Riemann—Cartan manifolds are distinguished in an invariant way. Geometric characteristics of some of the distinguished classes of Riemann—Cartan manifolds are found, and also conditions hindering the existence, are determined. The local geometry of Riemann—Cartan manifolds carrying pseudo-Killing and pseudoharmonic vector fields is studied. Conditions hindering the existence “in the large” of pseudo-Killing and pseudoharmonic vector fields on Riemann—Cartan manifolds are obtained.

Reply to ‘comment on ‘Braneworld remarks in Riemann–Cartan manifolds’’

In this brief reply, we elucidate some missing points in the comment (Khakshournia S 2009 Class. Quantum Grav. 26 178001) on our original paper (Hoff da Silva J M and da Rocha R 2009 Class. Quantum Grav. 26 055007), explicitly showing that the comment is unfounded in this context. We show that the term proposed equals zero, since the brane discontinuity is correctly defined in the torsion.

Comment on ‘Braneworld remarks in Riemann–Cartan manifolds’

In a recent paper (J M Hoff da Silva and da Rocha R 2009 Class. Quantum Grav. 26 055007) it is concluded that the Darmois–Israel junction conditions in the presence of torsion are not modified. We point out that this conclusion is invalid.

Braneworld remarks in Riemann–Cartan manifolds

We analyze the projected effective Einstein equation in a four-dimensional arbitrary manifold embedded in a five-dimensional Riemann–Cartan manifold. The Israel–Darmois matching conditions are investigated, in the context where the torsion discontinuity is orthogonal to the brane. Unexpectedly, the presence of torsion terms in the connection does not modify such conditions whatsoever, despite the modification in the extrinsic curvature and the connection. Then, by imposing the -symmetry, the Einstein equation obtained via Gauss–Codazzi formalism is extended in order to now encompass the torsion terms. We also show that the factors involving contorsion change drastically the effective Einstein equation on the brane, as well as the effective cosmological constant.

Nonholonomic versus vakonomic dynamics on a Riemann–Cartan manifold

For the Chaplygin’s nonholonomic constrained systems, the constraint manifold can be endowed with Riemann–Cartan geometric structure by nonholonomic mapping into a Riemann manifold. The two kinds of existing dynamics, nonholonomic dynamics and vakonomic dynamics, are compared in the framework of Riemann–Cartan geometry. It is proved that the equations of motion for nonholonomic and vakonomic dynamics are described by the equations of autoparallel and geodesic trajectories on the Riemann–Cartan constraint manifold, respectively. If the metricity condition of Riemann–Cartan connection is satisfied, the torsion (contorsion) of the Riemann–Cartan manifold characterizes the difference between the autoparallel and geodesic trajectories as well as the distinction between the nonholonomic and vakonomic equations.

The Topological Quantization and Bifurcation of Strings in Early Universe**The project supported by National Natural Science Foundation of China

We present a new topological invariant to describe the spacetime defect in Riemann–Cartan manifold. By means of the -mapping theory and the rank-two topological tensor current (associated to the new topological invariant), we show that there must exist many string objects which naturally generated from the zero points of -mapping and is quantized in the topological level under the condition that the Jacobian . When , we find that there exists a crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the origin and bifurcation of the strings are detailed in the neighborhoods of the limit points and bifurcation points of -mapping, respectively.

THE QUANTIZATION AND PRODUCTION OF THE SPACE–TIME DEFECTS IN THE EARLY UNIVERSE

In Riemann–Cartan manifold U4, a new topological invariant is obtained by means of the torsion tensor. In order to describe the space–time defects (which appear in the early universe due to torsion) in an invariant form, the new topological invariant is introduced to measure the size of defects and it is interpreted as the dislocation flux in internal space. By the use of the so-called ϕ mapping method and the gauge potential decomposition, the dislocation flux is quantized in units of the Planck length. The quantum numbers are determined by the Hopf indices and the Brouwer degrees. Furthermore, the dynamic form of the dislocations is studied by defining an identically conserved current. Based on the implicit function theorem, the production of the defects at the limit points is detailed.