Cartan Space Research Articles

On model mutation for reductive Cartan geometries and non-existence of Cartan space forms

Reductive models $((\frak g,\frak h),H)$ for Cartan geometries are showed to fall into two classes, symmetric and non symmetric type, according to the existence or non existence of a mutation $\frak g'=\frak h\oplus\frak m$ where the $H$-module $\frak m$ is an abelian subalgebra. Sasakian structures are showed to be Cartan geometries having a model of non symmetric type and other examples of models of this type are exhibited. Reductive models for which no Cartan space forms exist are constructed. The phenomenon of non-existence of Cartan space forms pertains to models of non symmetric type.

The geometry of Hamilton spaces - An introduction

A short introduction of Hamilton, Cartan and Generalized Hamilton spaces is provided. The Hamilton--Jacobi equations are deduced by variational calculus.

Cosmic Strings and Quintessence

Using torsion two-form we present a new Lorentz gauge invariant U(1)topological field theory in Riemann–Cartan space–time manifold U4.By virtue of the decomposition theory of U(1) gauge potential and theϕ-mapping topological current theory, it is proven that theU(1) complex scalar field ϕ(x) can be looked upon as theorder parameter field in our Universe, and a set of zero pointsof ϕ(x) create the cosmic strings as the space–time defectsin the early Universe. In the standard cosmology, this complexscalar order parameter field possesses negative pressure, providesan accelerating expansion of Universe, and be able to explain theinflation in the early Universe. Therefore this complex scalarfield is not only the order parameter field created the cosmicstrings in the early universe, but also reasonably behaves as the quintessence, the darkenergy.

GRAVITATIONAL CONSTANT AND TORSION

Riemann–Cartan space–time U4 is considered here. It has been shown that when we link topological Nieh–Yan density with the gravitational constant, we then obtain Einstein–Hilbert Lagrangian as a consequence.

Spinorial Matter in Affine Theory of Gravity and the Space Problem

The purely affine theory of gravity possesses a canonical formulation. For this and other reasons, it could be a promising candidate for quantum gravity. Motivated by these perspectives, we discuss spinorial matter coupled to gravity, where the latter is described by a connection having no a priori relation to a metric. We show that one can establish a truncated spinor formalism which, for special or approximate solutions to the gravitational equations, reduces to the standard formalism. As a consequence, one arrives at "matter-induced" Riemann–Cartan spaces solving the Weyl-Cartan space problem.

The Einstein 3-form Gα and its Equivalent 1-form Lα in Riemann–Cartan Space

The definition of the Einstein 3-form G_a is motivated by means of the contracted 2nd Bianchi identity. This definition involves at first the complete curvature 2-form. The 1-form L_a is defined via G_a = L^b \wedge #(o_b \wedge o_a). Here # denotes the Hodge-star, o_a the coframe, and \wedge the exterior product. The L_a is equivalent to the Einstein 3-form and represents a certain contraction of the curvature 2-form. A variational formula of Salgado on quadratic invariants of the L_a 1-form is discussed, generalized, and put into proper perspective.

Generally Covariant Schrödinger Equation in Newton–Cartan Space–Time. Part I

The covariant Schrodinger equation is obtained with the use of standard geometrical objects of the Galilean space–time. It's symmetry and covariance are investigated. Gauge freedom is eliminated by invariance condition. Family of the plane wave solutions in any coordinate system is found. Connection with previous investigations is discussed.

On inner tensor structures in Cartan spaces

Cartan spaces equipped with almost complex and almost antiquaternion structures are considered. The Hermitian metrics of almost complex Cartan spaces are found. It is proved that in the Cartan spaces there always exist Kahlerian metrics. The conditions of the complete integrability of these structures are established, and the method of constructing affine connections consistent with these structures is given.

The origin and bifurcation of dislocations in the gauge field theory of dislocation and disclination continuum

In 3-dimensional Riemann–Cartan space, the dynamic form of dislocation and the topological quantization of Burgers vector projection are obtained by extending the dislocation density projection and Burgers vector projection to a topological current. When the tangent stress inside the material body and order parameters of dislocation change, the origin and bifurcation of dislocations are studied. The branch solutions at the limit point and the different directions of all branch curves at the bifurcation point are calculated. Since the dislocation current is identically conserved, the total topological quantum numbers of the branched dislocations will remain constant, which is just the conservation law of Burgers vector in dislocation and disclination continuum. It is pointed out that a dislocation with a higher value of the Burgers vector is unstable, and it will evolve to the lower value of Burgers vector through a bifurcation process. Furthermore, one can see that the origin and bifurcation of dislocations are not gradual changes but sudden changes with the varying of tangent stress.

Curvature-free asymmetric metric connections and Lanczos potentials in Kerr–Schild spacetimes

The set of spinor equations linking the curvature spinors of a Riemann–Cartan space with the curvature spinors of the corresponding Riemann space is derived, and these are used to establish a simple relationship between Ψ-flat connections of the Riemann–Cartan space and Lanczos potentials of the Riemann space. This not only yields, very easily, a recent result of Bergqvist for the Kerr metric, but also enables Bergqvist’s result to be generalized; specifically we show that a curvature-free connection, associated with a class of Kerr–Schild metrics, can be identified as a Lanczos potential for the Weyl conformal curvature spinor of these spaces.

PERFECT FLUID AND TEST PARTICLE WITH SPIN AND DILATONIC CHARGE IN A WEYL–CARTAN SPACE

The equation of perfect dilaton-spin fluid motion in the form of generalized hydrodynamic Euler-type equation in a Weyl–Cartan space is derived. The isoentropic character of perfect dilaton-spin fluid motion is established. The equation of motion of a test particle with spin and dilatonic charge in the Weyl–Cartan geomtry background is obtained. The peculiarities of test particle motion in a Weyl–Cartan space are discussed.

Evaluation of the heat kernel in Riemann - Cartan space using the covariant Taylor expansion method

In order to evaluate the asymptotic expansion coefficients of the heat kernel associated with a fermion of spin in an arbitrary dimensional Riemann - Cartan spacetime, the limit of application of the covariant Taylor expansion method (Avramidi I G 1989 Teor. Mat. Fiz. 79 219) to the heat kernel is extended. The first three coefficients in the approach are manifestly computed. In the case of a massless fermion of spin coupled to a vector gauge field in six-dimensional Riemann - Cartan spacetime, the third coefficient which is intimately connected with some anomalies is expressed in the form of arranging it with respect to an irreducible -matrix basis.

Gauss–Bonnet Type Identity in Weyl–Cartan Space

The Gauss–Bonnet type identity is derived in a Weyl–Cartan space on the basis of the variational method.

Pontryagin, Euler Forms and Chern–Simons Terms in Weyl–Cartan Space

The existence of the Pontryagin and Euler forms in a Weyl–Cartan space on the basis of the variational method with Lagrange multipliers are established. It is proved that these forms can be expressed via the exterior derivatives of the corresponding Chern–Simons terms in a Weyl–Cartan space with torsion and nonmetricity.

The moduli space and monodromies of N = 2 supersymmetric SO(2 r + 1) Yang-Mills theory

We write down the weak-coupling limit of N = 2 supersymmetric Yang-Mills theory with arbitrary gauge group G. We find the weak-coupling monodromies represented in terms of Sp(2r, Z ) matrices depending on paths closed up to Weyl transformations in the Cartan space of complex dimension r, the rank of the group. There is a one to one relation between Weyl orbits of these paths and elements of a generalized braid group defined from G. We check that these weak-coupling monodromies behave correctly in limits of the moduli space corresponding to restrictions to subgroups. In the case of SO(2 r + 1) we write down the complex curve representing the solution of the theory. We show that the curve has the correct monodromies.

Hamilton geometry

The cotangent bundle T ∗ M of a finite-dimensional manifold M equipped with a regular Hamiltonian is studied. The notions of the metric d-tensor field, the canonical nonlinear connection and a metrical d-connection on T ∗ M are introduced in much the same way as in the already known case of the tangent bundle TM equipped with a regular Lagrangian. The relationship between the corresponding objects on T ∗ M and TM is via the Legendre transformation. The special case of a Cartan space is discussed and the so-called ecological metric (conformally equivalent to the m th root metric) is considered as an example.

Torsion, superconductivity, and massive electrodynamics

A model for superconductor, based on massive electrodynamics in spaces with torsion is given. The generalized London equations for Cartan spaces are presented. From the London equations we show that it is possible to obtain an expression for the magnetic permeability constant in vacuum in terms of the time component of the torsion vector.

On gauge-invariant and phase-invariant spinor analysis. II

Granted customary definitions, the operations of juggling indices and covariant differentiation do not commute with one another in a Weyl space. The same noncommutativity obtains in the spinor calculus of Infeld and van der Waerden. Gauge-invariant and phase-invariant calculations therefore tend to be rather cumbersome. Here, a modification of the definition of covariant derivative leads immediately to a manifestly gauge-invariant and phase-invariant version of Weyl–Cartan space and of the two-spinor calculus associated with it in which the metric tensor and the metric spinor are both covariant constant.

Lagrange multipliers and Gauss–Bonnet-type invariants in Riemann–Cartan space

A Lagrange multiplier approach is applied to Gauss–Bonnet-type invariants in Riemann–Cartan geometry. Lanczos-type identities are then derived. The use of the variational procedure in the context of gravitational theories with a Gauss–Bonnet invariant limit is discussed.

On the compatibility of relativistic wave equations in Riemann–Cartan spaces. II

In a Riemann–Cartan space‐time U4 the minimally coupled massive spin‐S Dirac equations are subject to constraints for all S> 1/2 . One may seek to modify these equations by the inclusion of appropriate supplementary terms α̂, β̂ (indices suppressed) in such a way that the modified equations are unconstrained. This is done here explicitly in the case where the field spinors are ξν1⋅⋅⋅νn and ημ̇ν1⋅⋅⋅νn−1, with n=2S. The spin‐1 tensor equations, i.e., the minimally coupled Proca equations, are transcribed into spinorial form. The explicit expressions for α̂ and β̂ that so appear are in harmony with those already obtained. The freedom to choose α̂ to be a zero spinor is shown to be circumscribed. Finally, it is pointed out that whether the unconstrained (spin‐1) equations are minimally coupled or not depends on whether one chooses to write them as tensor or spinor equations.