Torsion Tensor Research Articles

Normal connections on three-dimensional manifolds with solvable transformation group

The purpose of the work is the classification of three-dimensional homogeneous spaces, allowing a normal connection, description of invariant affine connections on those spaces together with their curvature and torsion tensors, holonomy algebras. We consider only the case, when Lie group is solvable. The local classification of homogeneous spaces is equivalent to the description of the effective pairs of Lie algebras. We study the holonomy algebras of homogeneous spaces and find when the invariant connection is normal. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces and they mainly have local character.

Spin precession of slow neutrons in Einstein-Cartan gravity with torsion, chameleon, and magnetic field

We analyse a spin precession of slow neutrons in the Einstein-Cartan gravity with torsion, chameleon and magnetic field. For the derivation of the Heisenberg equation of motion of the neutron spin we use the effective low-energy potential, derived by Ivanov and Wellenzohn (Phys. Rev. D92, 125004 (2015)) for slow neutrons, coupled to gravitational, chameleon and torsion fields to order 1/m, where m is the neutron mass. In addition to this low-energy interactions we switch on the interaction of slow neutrons with a magnetic field. We show that to linear order approximation with respect to gravitational, chameleon and torsion fields the Dirac Hamilton operator for fermions (neutrons), moving in spacetimes created by rotating coordinate systems, contains the anti-Hermitian operators of torsion-fermion (neutron) interactions, caused by torsion scalar and tensor degrees of freedom. Such anti-Hermitian operators violate CP and T invariance. One may assume that in the rotating Universe and galaxies the obtained anti-Hermitian torsion-fermion interactions might be an origin of i) violation of CP and T invariance in the Universe and ii) of baryon asymmetry.

Bol three-webs with covariantly constant curvature tensor

We find complete systemof tensor relations characterizing the class ofmultidimensional middle Bol three-webs with covariantly constant curvature tensor, and ascertain the algebraic sense of these relations. We prove the existence of such webs and lay the foundation of their classification in terms of torsion tensor rank. We also show that 6-dimensional non-group webs of such type are the known flexible webs E1 and E2.

The Einstein-Like Field Theory and Renormalization of the Shear Modulus

The Einstein-like field theory is developed to describe an elastic solid containing distribution of screw dislocations with finite-sized core. The core self-energy is given by a gauge-translational Lagrangian that is quadratic in torsion tensor and corresponding to the three-dimensional Riemann–Cartan geometry. The Hilbert–Einstein gauge equation plays the role of unconventional incompatibility law. The stress tensor of the modified screw dislocations is smoothed within the core. The renormalization of the shear modulus caused by proliferation of dipoles of nonsingular screw dislocations is studied. Bibliography: 23 titles.

The metric on field space, functional renormalization, and metric–torsion quantum gravity

Searching for new non-perturbatively renormalizable quantum gravity theories, functional renormalization group (RG) flows are studied on a theory space of action functionals depending on the metric and the torsion tensor, the latter parameterized by three irreducible component fields. A detailed comparison with Quantum Einstein–Cartan Gravity (QECG), Quantum Einstein Gravity (QEG), and “tetrad-only” gravity, all based on different theory spaces, is performed. It is demonstrated that, over a generic theory space, the construction of a functional RG equation (FRGE) for the effective average action requires the specification of a metric on the infinite-dimensional field manifold as an additional input. A modified FRGE is obtained if this metric is scale-dependent, as it happens in the metric–torsion system considered.

Geometric structures on solutions of equations of adiabatic gas motion

In this paper we show that characteristic covectors of equations of n-dimensional adiabatic gas motion, n = 1, 2, 3, generate a geometric structure on every their solution. This structure consists of a hyperplane and a non degenerate cone in each cotangent space to a solution so that the hyperplane and the cone intersect only at the zero point. We investigate differential invariants of this structure. In particular, we find a natural linear connection on every solution. A torsion tensor of this connection is trivial for n = 1. For n = 2, 3, this tensor is not trivial in general. For n = 1, we calculate solutions having the linear connection with zero curvature tensor. For n = 2, 3, we calculate solutions with zero torsion tensor.

Effective low-energy potential for slow Dirac fermions in Einstein-Cartan gravity with torsion and chameleon field

We derive the most general effective low-energy potential to order O(1/m) for slow Dirac fermions with mass m, coupled to gravitational, chameleon and torsion fields in the Einstein-Cartan gravity. The obtained results can be applied to the experimental analysis of gravitational, chameleon and torsion interactions in terrestrial laboratories. We discuss the use of rotating coordinate systems, caused by rotations of devices, for measurements of the torsion vector and tensor components, caused by minimal torsion--fermion couplings (Ivanov and Wellenzohn, Phys. Rev. D92, 065006 (2015)). Using the most general form of a metric tensor of curved spacetimes in rotating coordinate systems, proposed by Obukhov, Silenko, and Teryaev (Phys. Rev. D84, 024025 (2011)), we extend this metric by the inclusion of the chameleon field and calculate the set of vierbein fields, in terms of which Dirac fermions couple to torsion vector and tensor components through minimal torsion-fermion couplings. For such a set of vierbein fields we discuss a part of the effective low-energy potential for slow Dirac fermions, coupled to gravitational, chameleon and torsion fields to order O(1) in the large fermion mass expansion.

Berwald-type and Yano-type connections on Lie algebroids

In a new approach, by using the exact sequences, semisprays on the prolongation of a Lie algebroids are introduced and many important results on the semisprays and sprays are obtained. Also, the horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on the prolongation of a Lie algebroid are considered. Then the distinguished connections on the prolongation of a Lie algebroid are introduced and the torsion and curvature tensors of these connections are considered. In particular, the Berwald-type and Yano-type connections are studied.

The Weyl–Cartan Gauss–Bonnet gravity

In this paper, we consider the generalized Gauss–Bonnet action in four-dimensional Weyl–Cartan spacetime. In this spacetime, the presence of a torsion tensor and Weyl vector implies that the generalized Gauss–Bonnet action will not be a total derivative in four-dimensional spacetime. It will be shown that the higher than two time derivatives can be removed from the action by choosing a suitable set of parameters. In the special case where only the trace part of the torsion remains, the model reduces to general relativity plus two vector fields, one of which is massless and the other is massive. We will then obtain the healthy region of the five-dimensional parameter space of the theory in some special cases.

An introduction to the physics of Cartan gravity

A distance can be measured by monitoring how much a wheel has rotated when rolled without slipping. This simple idea underlies the mathematics of Cartan geometry. The Cartan-geometric description of gravity consists of a SO(1,4) gauge connection AAB(x) and a gravitational Higgs field VA(x) which breaks the gauge symmetry. The clear similarity with symmetry-broken Yang–Mills theory suggests strongly the existence of a new field VA in nature: the gravitational Higgs field. By treating VA as a genuine dynamical field we arrive at a natural generalization of General Relativity with a wealth of new phenomenology. Importantly, General Relativity is reproduced exactly in the limit that the SO(1,4) norm V2(x) tends to a positive constant. We show that in regions wherein V2 varies–but has a definite sign–the Cartan-geometric formulation is a particular version of a scalar–tensor theory (in the sense of gravity being described by a scalar field ϕ, metric tensor gμν, and possibly a torsion tensor Tμνρ). A specific choice of action yields the Peebles–Ratra quintessence model whilst more general actions are shown to exhibit propagation of torsion. Regions where the sign of V2 changes correspond to a change in signature of the geometry. Specifically, a simple choice of action with FRW symmetry imposed yields, without any additional ad hoc assumptions, a classical analogue of the Hartle–Hawking no-boundary proposal with the big bang singularity replaced by signature change. Cosmological solutions from more general actions are described, none of which have a big bang singularity, with most solutions reproducing General Relativity, or its Euclidean version, for late cosmological times. Requiring that gravity couples to matter fields through the gauge prescription forces a fundamental change in the description of bosonic matter fields: the equations of motion of all matter fields become first-order partial differential equations with the scalar and Dirac actions taking on structurally similar first-order forms. All matter actions reduce to the standard ones in the limit V2→const. We argue that Cartan geometry may function as a novel platform for inspiring and exploring modified theories of gravity with applications to dark energy, black holes, and early-universe cosmology. We end by listing a set of open problems.

Kerr-NUT black hole thermodynamics in f(T) gravity theories

An axially symmetric tetrad field is applied to the field equation of f(T) gravity theory, where T is the scalar torsion tensor. A solution, with three constants of integration, for the linear case, $ f(T)=T$ , is derived. Then, we substitute this solution in the tetrad field and change the angle $ \phi$ to $ \Phi$ , such that $ \Phi$ is a function of r , $ \theta$ and $ \phi$ , i.e. $ \Phi=N(r,\theta,\phi)$ . We calculate the scalar torsion and its derivatives, then, a constraint on the scalar torsion is assumed to be constant. From this assumption the value of $ N(r,\theta,\phi)$ is derived. We show that the value of $ N(r,\theta,\phi)$ is a solution to the f(T) gravitational theories provided some restrictions on the form of f(T) and its first derivatives. The associated metric of the derived solution is shown to represent the Kerr-NUT spacetime. The energy of the derived solution is calculated using the Euclidean continuation method and we show that the NUT parameter may be related to the electromagnetic field which is a consistent value with what was obtained before from other definitions. Finally, we calculate the thermodynamical quantities of the Kerr-NUT space times and investigate the first law of thermodynamics and quantum statistical relation.

The hidden flat like universe

We study a single fluid component in a flat like universe (FLU) governed by $f(T)$ gravity theories, where $T$ is the teleparallel torsion scalar. The FLU model, regardless the value of the spatial curvature $k$, identifies a special class of $f(T)$ gravity theories. Remarkably, the FLU $f(T)$ gravity does not reduce to teleparallel gravity theory. In large Hubble spacetime the theory is consistent with the inflationary universe scenario and respects the conservation principle. The equation of state (EoS) evolves similarly in all models $k=0, \pm 1$. We study the case when the torsion tensor is made of a scalar field, which enables to derive a quintessence potential from the obtained $f(T)$ gravity theory. The potential produces Starobinsky-like model naturally without using a conformal transformation, with higher orders continuously interpolate between Starobinsky and quadratic inflation models. The slow-roll analysis shows double solutions so that for a single value of the scalar tilt (spectral index) $n_{s}$ the theory can predict double tensor-to-scalar ratios $r$ of $E$-mode and $B$-mode polarizations.

Nearly Sasakian geometry and $$SU(2)$$ S U ( 2 ) -structures

We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or $5$-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the $5$-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo $SU(2)$-structures. By deforming such a $SU(2)$-structure one obtains in fact a Sasaki-Einstein structure. Further we prove that both nearly Sasakian and Sasaki-Einstein $5$-manifolds are endowed with supplementary nearly cosymplectic structures. We show that there is a one-to-one correspondence between nearly cosymplectic structures and a special class of hypo $SU(2)$-structures which is again strictly related to Sasaki-Einstein structures. Furthermore, we study the orientable hypersurfaces of a nearly K\"{a}hler 6-manifold and, in the last part of the paper, we define canonical connections for nearly Sasakian manifolds, which play a role similar to the Gray connection in the context of nearly K\"{a}hler geometry. In dimension $5$ we determine a connection which parallelizes all the nearly Sasakian $SU(2)$-structure as well as the torsion tensor field. An analogous result holds also for Sasaki-Einstein structures.

Multidimensional smooth loops with universal elasticity

Let be a universal (isotopically invariant) identity that is derived from the elasticity identity . One of the authors has previously shown that a) each local loop of dimension with identity (briefly, a loop ) is a smooth middle Bol loop of dimension ; b) smooth two-dimensional loops are Lie groups; c) up to isotopy, there exist only two three-dimensional loops : the loops and . In this paper, the loops and are extended to the multidimensional case. The fact that each smooth loop of dimension corresponds to a unique multidimensional three-web on a manifold of dimension is key to our work. In addition, the class of loops under investigation is characterized by the fact that the torsion tensor of the corresponding web has rank 1 (that is, the algebra generated by this tensor has a one-dimensional derived algebra). This enables us to express the differential equations of the problem in an invariant form. The system of equations thus obtained was found to be amenable to integration in the most general case, and the equations of the required loops have been obtained in local coordinates. Bibliography: 17 titles.

Kerr–Newman-NUT Black Hole in f(T) Gravity Theory and Its Thermodynamical Quantities

We show by explicit calculations that any general relativity solution remains a solution in f(T) gravitational theories provided some constraints on f(T) and its first derivatives are satisfied. This is done by applying an axially-symmetric tetrad field to the charged field equation of f(T), where \(T = T^{\mu }_{\nu \lambda }S_{\mu }^{\nu \lambda }\) is the scalar torsion tensor, gravity theory. In the linear case, f(T) = T which corresponds to general relativity theory, we derive solution with four constants of integration. For the non-linear case, we substitute the derived solution in the tetrad field and change the azimuthal coordinate to be a function of r, θ and ϕ, i.e., \(\Phi = L(r,\theta ,\phi )\). We calculate the scalar torsion and its derivatives then, a condition one the scalar torsion to be constant is assumed. We derive the value of \(L(r,\theta ,\phi )\) which satisfies this condition. The associated metric of the derived solution is shown to be a charged Kerr-NUT space time. The energy of this solution is calculated using the Euclidean continuation method and a consistent value is derived. We also calculate the thermodynamical quantities of charged Kerr-NUT space time and investigate the first law of thermodynamics and quantum statistical relation.

Weitzenböck’s torsion, Fermi coordinates, and adapted frames

We study Weitzenb\"ock's torsion and discuss its properties. Specifically, we calculate the measured components of Weitzenb\"ock's torsion tensor for a frame field adapted to static observers in a Fermi normal coordinate system that we establish along the world line of an arbitrary accelerated observer in general relativity. A similar calculation is carried out in the standard Schwarzschild-like coordinates for static observers in the exterior Kerr spacetime; we then compare our results with the corresponding curvature components. Our work supports the contention that in the extended general relativistic framework involving both the Levi-Civita and Weitzenb\"ock connections, curvature and torsion provide complementary representations of the gravitational field.

An Integrability Condition for Simple Lie Groups II

It is shown that a simple Lie group $G$ ($ \neq {\rm SL}_2$) can be locally characterised by an integrability condition on an $\operatorname{Aut}(\mathfrak{g})$ structure on the tangent bundle, where $\operatorname{Aut}(\mathfrak{g})$ is the automorphism group of the Lie algebra of $G$. The integrability condition is the vanishing of a torsion tensor of type $(1,2)$. This is a slight improvement of an earlier result proved in [Min-Oo M., Ruh E.A., in Differential Geometry and Complex Analysis, Springer, Berlin, 1985, 205-211].

SU(3)as a2-plectic manifold

The Killing form induces a 2-plectic structure on a compact semisimple Lie group. The associated Lie group of canonical transformations (2-plectomorphisms) is compact. This 2-plectic structure induces a Cartan connection on the Lie group. The curvature and torsion tensor of this connection have been calculated for the special unitary Lie group S U ( 3 ) . It is shown that the homogeneous spaces S U ( 3 ) S U ( 2 ) , S U ( 3 ) S 1 and S U ( 3 ) T 2 also, admit 2-plectic structures, which are induced by closed left invariant 3-forms on S U ( 3 ) , whereas S U ( 3 ) U ( 2 ) and S U ( 3 ) S O ( 3 ) do not admit such 2-plectic structures.

On Invariants of Almost Symplectic Connections

We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.

LIGHTLIKE HYPERSURFACES OF AN INDEFINITE KAEHLER MANIFOLD WITH A QUARTER-SYMMETRIC METRIC CONNECTION

Abstract. In this paper, we study lightlike hypersurfaces of an indefiniteKaehler manifold with a quarter-symmetric metric connection. We proveseveral classification theorems for such a lightlike hypersurface. 1. IntroductionA linear connection ∇¯ on a semi-Riemannian manifold (M,¯ ¯g) is said to bea quarter-symmetric connection if its torsion tensor T¯ satisfies(1.1) T¯(X,Y ) = π(Y )JX −π(X)JY,for any vector fields X and Y on M¯, where J is a (1,1)-type tensor field andπ is a 1-form associated with a non-vanishing smooth vector field ζ, which iscalled the torsion vector field of M¯, by π(X) = ¯g(X,ζ). Moreover, if ∇¯ satisfies∇¯¯g = 0, then it is called a quarter-symmetric metric connection.Quarter-symmetric metric connection was introduced by K. Yano and T.Imai [15], and then it have been studied by S. C. Rastogi [13, 14], D. Kamilyaand U. C. De [8], R. S. Mishra and S. N. Pandey [9], S. Golab [7] and others.On the other hand, N. Puˇsi´c [12], and J. Niki´c and Puˇsi´c [10] studied quarter-symmetric metric connections on Kaehler manifold.The theory of lightlike hypersurfaces is an important topic of research indifferential geometry due to its application in mathematical physics, especiallyin the general relativity. The study of such notion was initiated by Duggal andBejancu [3] and later studied by many authors (see recent results in two books[4, 6]). Although now we have lightlike version of a large variety of Riemann-ian submanifolds, the geometry of lightlike hypersurfaces of semi-Riemannianmanifolds with quarter-symmetric metric connections is hardly known.In this paper, we study lightlike hypersurfaces of an indefinite Kaehler man-ifold (M,¯ ¯g,J ) with a quarter-symmetric metric connection, in which the tensor