Compatible Connection Research Articles

An observation on Asanov's Unicorn metrics

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day. In this paper we prove that Asanov's Unicorns belong to the class of generalized Berwald manifolds. More precisely we prove the following theorems: a Finsleroid-Finsler space is a generalized Berwald space if and only if the Finsleroid charge is constant. Especially a Finsleroid-Finsler space is a Landsberg space if and only if it is a generalized Berwald manifold with a semi-symmetric compatible linear connection.

On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles

Let $(M,\sigma)$ be a compact Klein surface of genus $g \geq 2$ and let $E$ be a smooth Hermitian vector bundle on $M$. Let $\tau$ be a Real or Quaternionic structure on $E$ and denote respectively by $\mathcal{G}^{\tau}_{\mathbb{C}}$ and $\mathcal{G}^{\tau}_{E}$ the groups of complex linear and unitary automorphisms of $E$ that commute to $\tau$. In this paper, we study the action of $\mathcal{G}^{\tau}_{\mathbb{C}}$ on the space $\mathcal{A}^{\tau}_{E}$ of $\tau$-compatible unitary connections on $E$ and show that the closure of a semi-stable $\mathcal{G}^{\tau}_{\mathbb{C}}$-orbit contains a unique $\mathcal{G}^{\tau}_{E}$-orbit of projectively flat connections. We then use this invariant-theoretic perspective to prove a version of the Narasimhan–Seshadri correspondence in this context: $S$-equivalence classes of semi-stable Real and Quaternionic vector bundes are in bijective correspondence with equivalence classes of certain appropriate representations of orbifold fundamental groups of Real Seifert manifolds over the Klein surface $(M,\sigma)$.

Determination of Drugs for Crohn’s Disease Treatment in Pharmaceuticals by Capillary Electrophoresis Hyphenated with Tandem Mass Spectrometry

A capillary electrophoresis (CE) method hyphenated with tandem mass spectrometry (triple quadrupole, QqQ MS) was developed for the simultaneous determination of six selected drugs used in Crohn’s disease treatment, namely, azathioprine, 6-thioguanine, 6-mercaptopurine, mesalazine, prednisone, and allopurinol. A 10 mM ammonium acetate adjusted at pH 9 by 3% ammonium hydroxide and including a 5% methanol addition was used as an optimum background electrolyte (BGE). The optimum BGE provided both the baseline electrophoretic separation of the drugs and highly compatible connection of CE to the electro-spray ionization (ESI) interface. The optimized working conditions were favorable for the selective and sensitive QqQ MS detection of the separated compounds according to their mass-to-charge (m/z) ratios. The proposed method was validated in terms of precision (RSDs for the repeatability of migration times and peak areas of the analyzed drugs were less than 1.66 and 6.06%, respectively), linearity (determination coefficient ranged in the interval of 0.9987–0.9995), limits of quantitation (sub µg mL−1 levels), and accuracy (mean recoveries of all the analytes in pharmaceutical matrix ranged in the interval of 98.23–101.2%). The CE-ESI-QqQ MS method is fast, simple, selective, precise, and accurate, and was successfully applied to a highly reliable determination of the targeted drugs (for the treatment of inflammatory bowel disease, IBD) in commercial pharmaceuticals (tablet dosage forms).

The role of conformal symmetry in gravity and the standard model

In this paper we consider conformal symmetry in the context of manifolds with general affine connection. We extend the conformal transformation law of the metric to a general metric compatible affine connection, and find that it is a symmetry of both the geodesic equation and the Riemann tensor. We derive the generalised Jacobi equation and Raychaudhuri equation and show that they are both conformally invariant. Using the geodesic deviation (Jacobi) equation we analyse the behaviour of geodesics in different conformal frames. Since we find that our version of conformal symmetry is exact in classical pure Einstein's gravity, we ask whether one can extend it to the standard model. We find that it is possible to write conformal invariant Lagrangians in any dimensions for vector, fermion and scalar fields, but that such Lagrangians are only gauge invariant in four dimensions. Provided one introduces a dilaton field, gravity can be conformally coupled to matter.

An underlying geometrical manifold for Hamiltonian mechanics

We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture), that can be put into correspondence with the usual Hamilton–Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamiltonian–Lagrange picture, defined on the Hamilton–Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton–Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton–Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the the original Hamiltonian motion.

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

The “metric” structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e., compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e., the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case.

Notes on Conservation Laws, Equations of Motion of Matter, and Particle Fields in Lorentzian and Teleparallel de Sitter Space-Time Structures

We discuss the physics of interacting fields and particles living in a de Sitter Lorentzian manifold (dSLM), a submanifold of a 5-dimensional pseudo-Euclidean (5dPE) equipped with a metric tensor inherited from the metric of the 5dPE space. The dSLM is naturally oriented and time oriented and is the arena used to study the energy-momentum conservation law and equations of motion for physical systems living there. Two distinct de Sitter space-time structuresMdSLandMdSTPare introduced given dSLM, the first equipped with the Levi-Civita connection of its metric field and the second with a metric compatible parallel connection. Both connections are used only as mathematical devices. Thus, for example,MdSLis not supposed to be the model of any gravitational field in the General Relativity Theory (GRT). Misconceptions appearing in the literature concerning the motion of free particles in dSLM are clarified. Komar currents are introduced within Clifford bundle formalism permitting the presentation of Einstein equation as a Maxwell like equation and proving that in GRT there are infinitely many conserved currents. We prove that in GRT even when the appropriate Killing vector fields exist it is not possible to define a conserved energy-momentum covector as in special relativistic theories.

Gravitational energy for GR and Poincaré gauge theories: A covariant Hamiltonian approach

Our topic concerns a long standing puzzle: The energy of gravitating systems. More precisely we want to consider, for gravitating systems, how to best describe energy–momentum and angular momentum/center-of-mass momentum (CoMM). It is known that these quantities cannot be given by a local density. The modern understanding is that (i) they are quasi-local (associated with a closed 2-surface), (ii) they have no unique formula, (iii) they have no reference frame independent description. In the first part of this work, we review some early history, much of it not so well known, on the subject of gravitational energy in Einstein's general relativity (GR), noting especially Noether's contribution. In the second part, we review (including some new results) much of our covariant Hamiltonian formalism and apply it to Poincaré gauge theories of gravity (PG), with GR as a special case. The key point is that the Hamiltonian boundary term has two roles, it determines the quasi-local quantities, and furthermore, it determines the boundary conditions for the dynamical variables. Energy–momentum and angular momentum/CoMM are associated with the geometric symmetries under Poincaré transformations. They are best described in a local Poincaré gauge theory. The type of spacetime that naturally has this symmetry is Riemann–Cartan spacetime, with a metric compatible connection having, in general, both curvature and torsion. Thus our expression for the energy–momentum of physical systems is obtained via our covariant Hamiltonian formulation applied to the PG.

Self-adjointness of the Gaffney Laplacian on Vector Bundles

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.

On Randers manifolds with semi-symmetric compatible linear connections

Let M be a differentiable manifold equipped with a Riemannian metric tensor. If we apply translations in the tangent spaces to the Riemannian unit balls such that the translated bodies are keeping the origin in their interiors then they are working as unit balls for a Randers manifold. The unit balls allow us to measure the length of tangent vectors with the help of the induced Minkowski functionals. Analytically these Minkowski functionals are coming from a Riemannian metric tensor by using one-form perturbation in the tangent spaces. Manifolds equipped with a smoothly varying family of Minkowski functionals are called Finsler manifolds. The one-form perturbation of a Riemannian metric in the tangent spaces results in the class of Randers manifolds introduced by G. Randers in 1941. They occur naturally in physical applications related to electron optics, navigation problems or the Lagrangian of relativistic electrons. In this paper we are interested in Randers manifolds with semi-symmetric compatible linear connections. Compatibility means that the parallel transports preserve the length of tangent vectors with respect to the perturbed metric. If the torsion is decomposable in a special way then we speak about a semi-symmetric linear connection. Up to local isometries we characterize Riemannian manifolds admitting a one-form perturbation such that the resulting Randers manifold has a compatible semi-symmetric linear connection. As a paraphrase of our previous work (Vincze, 2006) communicated by Professor J.J. Duistermaat we present an existence theorem of generalized Berwald manifolds with semi-symmetric compatible linear connections. The terminology goes back to M. Matsumoto, M. Hashiguchi and S. Bácsó’s work (Bácsó et al., 1997).

Spinor and Twistor Geometry in Einstein Gravity and Finsler Modifications

We present a generalization of the spinor and twistor geometry for on (pseudo) Riemannian manifolds enabled with nonholonomic distributions or for Finsler-Cartan spaces modelled on tangent Lorentz bundles. Nonholonomic (Finsler) twistors are defined as solutions of generalized twistor equations determined by spin connections and frames adapted to nonlinear connection structures. We show that the constructions for local twistors can be globalized using nonholonomic deformations with "auxiliary" metric compatible connections completely determined by the metric structure and/or the Finsler fundamental function. We explain how to perform such an approach in the Einstein gravity theory formulated in Finsler like variables with conventional nonholonomic 2+2 splitting.

Leakage evaluation of original and compatible implant-abutment connections: In vitro study using Rhodamine B.

Leakage has been addressed as a major contributing factor to inflammatory reactions at the implant–abutment connection, leading to problems such as oral malodor, inflammation, and marginal bone loss. The aim of this study was to investigate in vitro the leakage at implant–abutment interface of OsseoSpeed™ implants connected to original and compatible abutments. A total of 28 OsseoSpeed implants were divided into four groups (n = 7). Each group was connected to four different abutments according to manufacturers’ recommendations: group A (TiDesign™); group B (Natea™); group C (Dual™); and group D (Implanet™) abutments. The inner volume of each implant–abutment combination was calculated and leakage was detected for each group with spectrophotometric analysis at 1 h (D0) and 48 h (D1) of incubation time using Rhodamine B. At 1 h, leakage volume was significantly lower in TiDesign and Dual than in Natea and Implanet (P < 0.001). At 48 h, however, leakage was significantly lower between TiDesign and all other systems (P < 0.005). Compatible abutments do not fit internal connection of OsseoSpeed implants perfectly, which increases the leakage of the final assembly.

Inadvertent intravenous administration of maternal breast milk in a six-week-old infant: a case report and review of the literature

BackgroundAccidental intravenous administration of an enteral feeding can be fatal or cause complications such as sepsis, acute respiratory and circulatory failure, acute renal failure, hepatic insufficiency, coagulation disorders and severe permanent neurological sequelae. These “wrong route” errors are possible due to compatible connections between enteral feeding systems and intravascular infusion catheters.Case presentationWe report a six-week-old male infant who received a 5 ml intravenous infusion of breast milk. Within five minutes of administration the child developed tachycardia and tachypnea, accompanied by a sudden decrease in oxygen saturation on pulse oximetry to 69%. The infant received supplemental oxygen via nasal cannula and was transferred to the pediatric intensive care unit. Broad-spectrum antibiotics were administered for 48 hours. Vital signs returned to normal within a few hours. Neurological follow-up through 3 years did not reveal any neurodevelopmental abnormalities.ConclusionDevelopment of specific enteral feeding connections, which are incompatible with intravascular catheter connections, is needed urgently to prevent a misconnection with potential morbidity or mortality of children.

On generalized Berwald manifolds with semi-symmetric compatible linear connections

On generalized Berwald manifolds with semi-symmetric compatible linear connections

On generalized Berwald manifolds with semi-symmetric compatible linear connections

On generalized Berwald manifolds with semi-symmetric compatible linear connections

Noncommutative Riemannian geometry on graphs

We show that arising out of noncommutative geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a ‘finite manifold structure’ on the set of vertices. We equip any graph with a canonical ‘Euclidean metric’ and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.

Incompatibility and preferred morphology in the self-accommodation microstructure of β-titanium shape memory alloy

The frequency distribution of habit plane variant (HPV) clusters and the deviation from twin orientation relationships (ORs) at the junction plane (JP) are investigated by transmission electron microscopy together with theoretical evaluation of the kinematic compatibility (KC) at the JP in a β-titanium shape memory alloy. Even though there are more than 10 types of possible HPV clusters, only three types are formed. V-shaped couplings of HPVs by {111} type I twins (VI: 49%) and by ⟨211⟩ type II twins (VII: 42%) are the predominant types. A triangular morphology due to coupling of {111} type I twins is observed with a frequency of only 9%. These preferred morphologies are well explained by the degree of incompatibility (the rotation necessary for compatible connection of HPVs). The exact twin OR and KC are maintained at the JP in a VI cluster instead of KC at the habit plane (HP), whereas the JP in a VII cluster is incompatible and the ⟨211⟩ type II twin OR shows slight deviation at the JP by about 0.4°. The competition between KC at the JP and KC at the HP (invariant plane) is responsible for the frequency distribution of HPV clusters and the character of the interfaces in the self-accommodation microstructure.

The odd side of torsion geometry

We introduce and study a notion of ‘Sasaki with torsion structure’ (st ) as an odd-dimensional analogue of Kähler with torsion geometry (kt ). These are normal almost contact metric manifolds that admit a unique compatible connection with \( 3 \)-form torsion. Any odd-dimensional compact Lie group is shown to admit such a structure; in this case, the structure is left-invariant and has closed torsion form. We illustrate the relation between st structures and other generalisations of Sasaki geometry, and we explain how some standard constructions in Sasaki geometry can be adapted to this setting. In particular, we relate the st structure to a kt structure on the space of leaves and show that both the cylinder and the cone over an st manifold are kt , although only the cylinder behaves well with respect to closedness of the torsion form. Finally, we introduce a notion of ‘\( G \)-moment map’. We provide criteria based on equivariant cohomology ensuring the existence of these maps and then apply them as a tool for reducing st structures.

Poisson bracket on 1-forms and evolutionary partial differential equations

We introduce a bracket on 1-forms defined on , i.e. the infinite jet extension of the space of loops, and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of F-manifolds with a compatible flat connection (M, ∇, ○) are Hamiltonian in a generalized sense. Moreover, we show that if a metric g compatible with ∇ is also invariant with respect to ○, then this generalized Hamiltonian setup reduces to the standard one.

PRINCIPLES OF EINSTEIN–FINSLER GRAVITY AND PERSPECTIVES IN MODERN COSMOLOGY

We study the geometric and physical foundations of Finsler gravity theories with metric compatible connections defined on tangent bundles, or (pseudo) Riemannian manifolds, endowed with nonholonomic frame structure. Several generalizations and alternatives to Einstein gravity are considered, including modifications with broken local Lorentz invariance. It is also shown how such theories (and general relativity) can be equivalently re-formulated in Finsler like variables. We focus on prospects in modern cosmology and Finsler acceleration of Universe. Einstein–Finsler gravity theories are elaborated following almost the same principles as in the general relativity theory but extended to Finsler metrics and connections. Finally, some examples of generic off-diagonal metrics and generalized connections, defining anisotropic cosmological Einstein–Finsler spaces are analyzed; certain criteria for the Finsler accelerating evolution are formulated.