2010 S. Vacaru's publications reviewed in Zentralblatt

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Zbl 1203.37106Vacaru, Sergiu I .Curve flow s and solitonic hierarchi es generated by Einstein metrics. (English)[J] Acta Appl. Math. 110, No. 1, 73-107 (2010). ISSN 0167-8019; ISSN 1572-9036

Using a pseudo-Riemannian metric and non-linear connections on a non-integrable distribution ona manifold, some bi-Hamiltonian structures and mKdV hierarchies of solitonic equations arestudied. Using a non-holonomical splitting, couples of generalized sine-Gordon equations areobtained. Following the geometric construction in the case of the Levi-Civita connection, one get a``nonholonomic mixing'' of solitonic interaction. A generous discussion for further research and

developments, as well as a technical Appendix are finally given.[Paul Popescu (Craiova) ]

MSC 2000:*37K05 Hamiltonian structures, etc.37K10 Completely integrable systems etc.37K25 Relations with differential geometry35Q53 KdV-like equations53B20 Local Riemannian geometry53B40 Finsler spaces and generalizations (local)53C21 Methods of Riemannian geometry (global)53C60 Finsler spaces and generalizations (global)

Keywords: bi-Hamiltonian structure; mKdV hierarchies; solitonic equations; (pseudo) Riemannianmetric; $N$-connection

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Zbl 1201.83024Vacaru, Sergiu I .Tw o-connection renormalization and non-holonomic gauge models of Einstein gravity.(English)[J] Int. J. Geom. Methods Mod. Phys. 7, No. 5, 713-744 (2010). ISSN 0219-8878

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Summary: A new framework for perturbative quantum gravity is proposed following the geometryof non-holonomic distributions on (pseudo)Riemannian manifolds. There are considered suchdistributions and adapted connections, also completely defined by a metric structure, whengravitational models with infinite many couplings reduce to two-loop renormalizable effectiveactions. We use a key result from our partner work arXiv:0902.0911 from 2009 that the classicalEinstein gravity theory can be reformulated equivalently as a non-holonomic gauge model in thebundle of affine/de Sitter frames on pseudo-Riemannian space-time. It is proven that (for a classof non-holonomic constraints and splitting of the Levi-Civita connection into a `renormalizable"distinguished connection, on a base background manifold, and a gauge-like distortion tensor, intotal space) a non-holonomic differential renormalization procedure for quantum gravitationalfields can be elaborated. Calculation labor is reduced to one- and two-loop levels andrenormalization group equations for non-holonomic configurations.

MSC 2000:*83C45 Quantization of the gravitational field58A30 Vector distributions (global analysis)53C50 Lorentz manifolds, manifolds with indefinite metrics53C29 Issues of holonomy83C05 Einstein's equations81T17 Renormalization group methods83D05 Relativistic gravitational theories other than Einstein's

Keywords: perturbative quantum gravity; non-holonomic manifolds; nonlinear connections;Einstein gravity; gauge gravity


Zbl 1200.83060Anastasiei, Mihai ; Vacaru, Sergiu I .Nonholonomic black ring and solitonic solutions in F insler and extra dimension gravitytheories. (English)[J] Int. J. Theor. Phys. 49, No. 8, 1788-1804 (2010). ISSN 0020-7748; ISSN 1572-9575

Summary: We study stationary configurations mimicking nonholonomic locally anisotropic blackrings (for instance, with ellipsoidal polarizations and/or imbedded into solitonic backgrounds) inthree/six dimensional pseudo-Finsler/Riemannian spacetimes. In the asymptotically flat limit, forholonomic configurations, a subclass of such spacetimes contains the set of five dimensional blackring solutions with regular rotating event horizon. For corresponding parameterizations, themetrics and connections define Finsler-Einstein geometries modeled on tangent bundles, or onnonholonomic (pseudo) Riemannian manifolds. In general, there are vacuum nonholonomicgravitational configurations which can not be generated in the limit of zero cosmological constant.

MSC 2000:*83C57 Black holes83E15 Higher-dimensional field theories83C15 Closed form solutions of equations in general relativity83D05 Relativistic gravitational theories other than Einstein's53B40 Finsler spaces and generalizations (local)

Keywords: pseudo-Finsler geometry; nonholonomic manifolds and bundles; nonlinear connections;black rings


Zbl 1196.83014Vacaru, Sergiu I .Einstein gravity in almos t Kähler and Lagrange-Finsler variables and deformationquantization. (English)


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[J] J. Geom. Phys. 60, No. 10, 1289-1305 (2010). ISSN 0393-0440

Summary: A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics andconnections into canonical geometric objects (metric and nonlinear and linear connections) foreffective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented asalmost Kähler spaces. This allows us to formulate an approach to quantum gravity followingstandard methods of deformation quantization. Such constructions are performed not on tangentbundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributionsdefining $2+2$ splitting with associate nonlinear connection structure. We also show how theEinstein equations can be written in terms of Lagrange-Finsler variables and corresponding almostsymplectic structures and encoded into the zero-degree cohomology coefficient for a quantummodel of Einstein manifolds.

MSC 2000:*83C05 Einstein's equations53D55 Deformation quantization, star products53B40 Finsler spaces and generalizations (local)53B35 Complex differential geometry (local)83C45 Quantization of the gravitational field

Keywords: Einstein spaces; Lagrange geometry; Finsler geometry; deformation quantization;

quantum gravityPDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

Zbl 1194.58002Vacaru, Sergiu I .Nonholonomic distributions and gauge models of Einstein gravity. (English)[J] Int. J. Geom. Methods Mod. Phys. 7, No. 2, 215-246 (2010). ISSN 0219-8878

Pseudo-Riemannian metric and a compatible linear connection are studied in the presence of $2 +2$ nonholonomic distribution preserved by the parallel transports of the connection. The authordescribes the geometric structures on a 4-manifold giving this triple and reformulates the Einsteinequations in nonholonomic coordinates (special tetrad formalism). It is shown how the Einsteingravity theory can be redefined equivalently as certain gauge models on nonholonomic affine orde Sitter frame bundles.[Boris S. Kruglikov (Troms\o) ]

MSC 2000:*58A30 Vector distributions (global analysis)35Q7653C07 Special connections and metrics on vector bundles83C05 Einstein's equations

Keywords: nonholonomic distributions; nonlinear connections; Einstein equations; gauge gravity


Zbl 1190.83067Vacaru, Sergiu I .Finsler black holes induced by no ncommutative anholonomic distributions in Einsteingravity. (English)[J] Classical Quantum Gravity 27, No. 10, Article ID 105003, 19 p. (2010). ISSN 0264-9381;ISSN 1361-6382

Summary: We study Finsler black holes induced from Einstein gravity as possible effects of quantum spacetime noncommutativity. Such Finsler models are defined by nonholonomic framesnot on tangent bundles but on (pseudo)Riemannian manifolds being compatible with standard


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theories of physics. We focus on noncommutative deformations of Schwarzschild metrics intolocally anisotropic stationary ones with spherical/rotoid symmetry. The conditions are derivedwhen black hole configurations can be extracted from two classes of exact solutions depending onnoncommutative parameters. The first class of metrics is defined by nonholonomic deformations of the gravitational vacuum by noncommutative geometry. The second class of such solutions isinduced by noncommutative matter fields and/or effective polarizations of cosmological constants.

MSC 2000:*83C57 Black holes

83C65 Methods of noncommutative geometry83C45 Quantization of the gravitational field53B40 Finsler spaces and generalizations (local)53C29 Issues of holonomy


Zbl 1190.83030Vacaru, Sergiu I .On general solutions for field equations in Einstein and higher dimension gravity.(English)[J] Int. J. Theor. Phys. 49, No. 4, 884-913 (2010). ISSN 0020-7748; ISSN 1572-9575

Summary: We prove that the Einstein equations can be solved in a very general form for arbitraryspacetime dimensions and various types of vacuum and non-vacuum cases following a geometricmethod of anholonomic frame deformations for constructing exact solutions in gravity. The mainidea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to theLevi-Civita connection) metric compatible linear connection which is also completely defined by thesame metric structure. Such a canonically distinguished connection is with nontrivial torsion whichis induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It ispossible to define certain classes of adapted frames of reference when the Einstein equations forsuch an alternative connection transform into a system of partial differential equations which canbe integrated in very general forms. Imposing nonholonomic constraints on generalized metricsand connections and adapted frames (selecting Levi-Civita configurations), we generate exact

solutions in Einstein gravity and extra dimension generalizations.MSC 2000:

*83C15 Closed form solutions of equations in general relativity83E15 Higher-dimensional field theories

Keywords: Einstein spaces and higher dimension gravity; anholonomic frames; exact solutions;nonholonomic manifolds


Zbl pre05828781Vacaru, Sergiu I .New classes of off-diagonal cosmological solutions in Einstein gravity. (English)[J] Int. J. Theor. Phys. 49, No. 11, 2753-2776 (2010). ISSN 0020-7748; ISSN 1572-9575

Summary: In this work, we apply the anholonomic deformation method for constructing newclasses of anisotropic cosmological solutions in Einstein gravity and/or generalizations withnonholonomic variables. There are analyzed four types of, in general, inhomogeneous metrics,defined with respect to anholonomic frames and their main geometric properties. Such spacetimescontain as particular cases certain conformal and/or frame transforms of the well knownFriedman-Robertson-Walker, Bianchi, Kasner and Gödel universes and define a great variety of cosmological models with generic off-diagonal metrics, local anisotropy and inhomogeneity. It isshown that certain nonholonomic gravitational configurations may mimic de Sitter like inflation


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scenarios and different anisotropic modifications without satisfying any classical false-vacuumequation of state. Finally, we speculate on perspectives when such off-diagonal solutions can berelated to dark energy and dark matter problems in modern cosmology.

MSC 2000:*83F05 Relativistic cosmology83C15 Closed form solutions of equations in general relativity83C55 Hydrodynamics (general relativity)

Keywords: anisotropic cosmology; off-diagonal metrics; exact solutions in gravity; nonholonomicdeformations


Zbl 1200.53082Anastasiei, Mihai ; Vacaru, Sergiu I .Fedoso v quantization of Lagrange-Finsler and Hami lton-Cartan spaces and Einsteingravity lifts on (co) tangent bundles. (English)[J] J. Math. Phys. 50, No. 1, Paper No. 013510, 23 p. (2009). ISSN 0022-2488

Summary: We provide a method of converting Lagrange and Finsler spaces and their Legendre

transforms to Hamilton and Cartan spaces into almost Kähler structures on tangent and cotangentbundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo)Riemannian / Einstein metrics on effective phase spaces. This allows us to define thecorresponding Fedosov operators and develop deformation quantization schemes for nonlinearmechanical and gravity models on Lagrange- and Hamilton-Fedosov manifolds. Editorial remark:No review copy delivered.

MSC 2000:*53D55 Deformation quantization, star products53C60 Finsler spaces and generalizations (global)81S10 Geometric quantization, symplectic methods

Keywords: geometry; quantisation (quantum theory); quantum gravity

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Zbl 1183.83045Vacaru, Sergiu I .Branes and quantization for an A-model complexificatio n of Einstein gravity in almostKähler variables. (English)[J] Int. J. Geom. Methods Mod. Phys. 6, No. 6, 873-909 (2009). ISSN 0219-8878

Author's abstract: The general relativity theory is redefined equivalently in almost Kählervariables: symplectic form and canonical symplectic connection (distorted from the Levi-Civita

connection by a tensor constructed only from metric coefficients and their derivatives). Thefundamental geometric and physical objects are uniquely determined in metric compatible form bya (pseudo) Riemannian metric on a manifold enabled with a necessary type nonholonomic 2+2distribution. Such nonholonomic symplectic variables allow us to formulate the problem of quantizing Einstein gravity in terms of the A-model complexification of almost complex structureson spacetime manifold, generalizing the Gukov-Witten method [see {\it S. Gukov, E. Witten},Branes and Quantization, arXiv:0809.0305 ]. Quantizing the complexified model, we derive aHilbert space as a space of strings with two A-branes which for the Einstein gravity theory arenonholonomic because of induced nonlinear connection structures. Finally, we speculate onrelation of such a method of quantization to curve flows and solitonic hierarchies defined byEinstein metrics on (pseudo) Riemannian spacetimes.[Benjamin Cahen (Metz) ]


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MSC 2000:*83C45 Quantization of the gravitational field81S10 Geometric quantization, symplectic methods53D55 Deformation quantization, star products53B40 Finsler spaces and generalizations (local)53B35 Complex differential geometry (local)53D50 Geometric quantization

Keywords: quantum gravity; Einstein gravity; nonholonomic manifolds; symplectic variables;

nonlinear connections; strings; A-branesPDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

Zbl 1175.83059Vacaru, Sergiu I .Einstein gravity in almost Kähler variables and stability of gravity w ith nonholo nomicdistributions and nonsymmetric metrics. (English)[J] Int. J. Theor. Phys. 48, No. 7, 1973-1999 (2009). ISSN 0020-7748; ISSN 1572-9575

Summary: We argue that the Einstein gravity theory can be reformulated in almost Kähler(nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely

defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation onmanifolds enabled with nonholonomic distributions is considered. We prove that, for certain typesof nonholonomic constraints, there are modelled effective Lagrangians which do not developinstabilities. It is also elaborated a linearization formalism for anholonomic noncommutativegravity theories models and analyzed the stability of stationary ellipsoidal solutions defining somenonholonomic and/or nonsymmetric deformations of the Schwarzschild metric. We show how toconstruct nonholonomic distributions which remove instabilities in nonsymmetric gravity theories.It is concluded that instabilities do not consist a general feature of theories of gravity withnonsymmetric metrics but a particular property of some models and/or unconstrained solutions.

MSC 2000:*83D05 Relativistic gravitational theories other than Einstein's83C05 Einstein's equations53B35 Complex differential geometry (local)

Keywords: gravity and symplectic variables; nonsymmetric metrics; nonholonomic manifolds;nonlinear connections; stability


Zbl 1173.53029Vacaru, Sergiu I .The entropy of Lagrange-Finsler spaces and Ri cci flow s. (English)[J] Rep. Math. Phys. 63, No. 1, 95-110 (2009). ISSN 0034-4877

The present paper is a possible extension of Perelman's approach to Ricci flows in order to derivethe evolution equations for Lagrange and Finsler geometries. Moreover, in the final part astatistical analogue of regular mechanical systems is discussed. The author is following quiteclosely the classical theory using for the extension the Levi-Civita connection of the Sasaki metricinduces by a Lagrange or Finsler structure on the tangent space. The concrete evolution equationsare computed in the paper. This seems to be one of the first attempts to apply the celebratedtheory of Ricci flows to more general manifolds than Riemannian ones. Even though the paper inthe present form contains merely the setting of such an attempt, one can expect furtherdevelopments of this approach that will clarify the topological properties of Lagrange andFinslerian structures.[Hideo Shimada (Sapporo) ]


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MSC 2000:*53C44 Geometric evolution equations (mean curvature flow)53C21 Methods of Riemannian geometry (global)53C25 Special Riemannian manifolds83C15 Closed form solutions of equations in general relativity83C99 General relativity83E99 Unified, higher-dimensional and super field theories53C60 Finsler spaces and generalizations (global)

Keywords: Ricci flows; nonholonomic manifolds; Lagrange geometry; Finsler geometry; nonlinearconnections


Zbl 1162.83359Vacaru, Sergiu I .Non holono mic Ricci flow s, exact soluti ons in gravity, and Symmetric and nonsymmetricmetrics. (English)[J] Int. J. Theor. Phys. 48, No. 2, 579-606 (2009). ISSN 0020-7748; ISSN 1572-9575

Summary: We provide a proof that nonholonomically constrained Ricci flows of (pseudo)

Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we considerflows of some physically valuable exact solutions in general relativity). There are constructed andanalyzed three classes of solutions of Ricci flow evolution equations defining nonholonomicdeformations of Taub NUT, Schwarzschild, solitonic and pp-wave symmetric metrics intononsymmetric ones.

MSC 2000:*83E05 Geometrodynamics83C47 Quantum field theory on curved space-times83C57 Black holes

Keywords: nonsymmetric metrics; nonholonomic manifolds; nonlinear connections; nonholonomicRicci flows; Taub NUT spacetimes; solitons in gravity; pp-Waves


Zbl 1153.37409A nco, Stephen C. ; Vacaru, Sergiu I .Curve flow s in Lagrange-Finsler geometry, bi-Hamiltoni an structures and solitons .(English)[J] J. Geom. Phys. 59, No. 1, 79-103 (2009). ISSN 0393-0440

Summary: Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonianstructures and related mKdV hierarchies of soliton equations derived geometrically from regularLagrangians and flows of non-stretching curves in tangent bundles. The total space geometry andnonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinearconnections ($N$-connections), Sasaki type metrics and linear connections. The simplest examplesof such geometries are given by tangent bundles on Riemannian symmetric spaces $G/SO(n)$provided with an $N$-connection structure and an adapted metric, for which we elaborate acomplete classification, and by generalized Lagrange spaces with constant Hessian. \par In thisapproach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo)Riemannian metrics in gravity. The results yield horizontal/vertical pairs of vector sine-Gordonequations and vector mKdV equations, with the corresponding geometric curve flows in thehierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrödinger maps on a tangent bundle.

MSC 2000:


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*37K05 Hamiltonian structures, etc.37K10 Completely integrable systems etc.37K25 Relations with differential geometry35Q53 KdV-like equations53B20 Local Riemannian geometry53B40 Finsler spaces and generalizations (local)53C21 Methods of Riemannian geometry (global)53C60 Finsler spaces and generalizations (global)

Keywords: curve flow; (semi) Riemannian spaces; nonholonomic manifold; nonlinear connection;Lagrange and Finsler geometry; bi-Hamiltonian; soliton equation


Zbl pre05840824Vacaru, Sergiu I .Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows.(English)[J] J. Math. Phys. 50, No. 7, 073503, 24 p. (2009). ISSN 0022-2488

Editorial remark: No review copy delivered

MSC 2000:*81-99 Quantum theory

Keywords: Dirac equation; geometry; noncommutative field theory


Zbl 1175.53034Vacaru, Sergiu I . ; Gonzáles-Hernández, Juan F.Non linear connections on gerbes, Clifford-Finsle r modules and the index theorems.(English)[J] Indian J. Math. 50, No. 3, 573-606 (2008). ISSN 0019-5324

The authors propose to develop the geometry of non-holonomic bundle gerbes having a nonlinearconnection structure, and non-holonomic gerbe modules, as a theory of Clifford modules onnon-holonomic manifolds, which need not necessarily be spin-manifolds. They considernon-holonomic Dirac operators and derive the related Atiyah-Singer index formulae. Their paperconcludes with certain applications in modern gravity and the geometric mechanics of Clifford-Lagrange/Finsler gerbes and their realization as non-holonomic Clifford and Riemann-Cartanmodules. Contents include: Introduction; $n$-anholonomic manifolds; Examples of $n$-anholonomic spaces; Lifts of non-holonomic bundle gerbes and connections; Non-holonomicClifford grebes and modules; Some local formulae from $n$-connection geometry; and References(fifty-seven items).[ Joseph D. Zund (Las Cruces) ]

MSC 2000:*53C0855R65 Generalizations of fiber spaces and bundles53C05 Connections, general theory53C27 Spin and Spin$^c$ geometry57R20 Characteristic classes and numbers57R22 Topology of vector bundles and fiber bundles53B20 Local Riemannian geometry70H99 Hamiltonian and Lagrangian mechanics81T13 Gauge theories83C60 Spinor and twistor methods in general retativity


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Keywords: non-holonomic bundle gerbes; Clifford modules; Dirac operator; Atiyah-Singer indexformula

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Zbl 1162.53329Vacaru, Sergiu I .Einstein gravity, Lagrange-Finsler geometry, and nonsymmetric metrics . (English)[J] SIGMA, Symmetry Integrability Geom. Methods Appl. 4, Paper 071, 29 p., electronic only(2008). ISSN 1815-0659

Summary: We formulate an approach to the geometry of Riemann-Cartan spaces provided withnonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducingeffective nonlinear and affine connections. Such geometries can be modelled by movingnonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart-Moffat and Finsler-Lagrange spaces with connections compatibleto a general nonsymmetric metric structure. Elaborating a metrization procedure for arbitrarydistinguished connections, we define the class of distinguished linear connections which arecompatible with the nonlinear connection and general nonsymmetric metric structures. Thenonsymmetric gravity theory is formulated in terms of metric compatible connections. Finally,there are constructed such nonholonomic deformations of geometric structures when the Einsteinand/or Lagrange-Finsler manifolds are transformed equivalently into spaces with generic localanisotropy induced by nonsymmetric metrics and generalized connections. We speculate onpossible applications of such geometric methods in Einstein and generalized theories of gravity,analogous gravity and geometric mechanics.

MSC 2000:*53Z05 Appl. of differential geometry to physics53B40 Finsler spaces and generalizations (local)53C21 Methods of Riemannian geometry (global)53C12 Foliations (differential geometry)53C44 Geometric evolution equations (mean curvature flow)58A30 Vector distributions (global analysis)

Keywords: nonsymmetric metrics; nonholonomic manifolds; nonlinear connections; Eisenhart-Lagrange spaces; generalized Riemann-Finsler geometry

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Zbl 1152.81623Vacaru, Sergiu I .Nonholonomic Ricci flow s. II: Evolution equations and dynamics. (English)[J] J. Math. Phys. 49, No. 4, 043504, 27 p. (2008). ISSN 0022-2488

Editorial remark: No review copy delivered.

MSC 2000:*53C44 Geometric evolution equations (mean curvature flow)37J60 Nonholonomic dynamical systems53C60 Finsler spaces and generalizations (global)


Zbl 1152.53016Vacaru, Sergiu I .Finsl er and Lagrange geometries in Einstein and string gravity. (English)


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[J] Int. J. Geom. Methods Mod. Phys. 5, No. 4, 473-511 (2008). ISSN 0219-8878

In this survey paper the author presents the current status of Finsler-Lagrange geometry andgeneralizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometerswell informed of general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. He also proposes a canonical schemewhen geometrical objects on a (pseudo) Riemannian space are non-holonomically deformed intogeneralized Lagrange, or Finsler, configurations on the same manifold. Such canonical transformsare defined by the coefficients of a prime metric and generated target spaces as Lagrangestructures, their models of almost Hermitian/Kähler, or non-holonomic Riemann spaces. Finally,the author considers some classes of exact solutions in string and Einstein gravity modelingLagrange-Finsler structures with solitonic pp-waves and speculates on their physical meaning.[R. Iordanescu (Bucureşti) ]

MSC 2000:*53B40 Finsler spaces and generalizations (local)53B50 Appl. of local differential geometry to physics53C21 Methods of Riemannian geometry (global)53C55 Complex differential geometry (global)83C15 Closed form solutions of equations in general relativity83E99 Unified, higher-dimensional and super field theories


Zbl 1199.83004Vacaru, Sergiu I .Generalized Lagrange transforms: Finsler geometry methods and deformationquantization of gravity. (English)[J] An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 53, Suppl., 327-342 (2007). ISSN1221-8421; ISSN 0041-9109

Summary: We propose a natural Fedosov type quantization of generalized Lagrange models andgravity theories with metrics lifted on tangent bundle, or extended to higher dimension, following

some stated geometric/physical conditions (for instance, nonholonomic and/or conformaltransforms to some physically important metrics or mapping into a gauge model). Suchgeneralized Lagrange transforms define canonical nonlinear connection, metric and linearconnection structures and model almost K\" ahler geometries with induced canonical symplecticstructure and compatible affine connection. The constructions are possible due to a synthesis of the nonlinear connection formalism developed in Finsler and Lagrange geometries anddeformation quantization methods.

MSC 2000:*83C45 Quantization of the gravitational field81S10 Geometric quantization, symplectic methods53D55 Deformation quantization, star products53B40 Finsler spaces and generalizations (local)53B35 Complex differential geometry (local)53D50 Geometric quantization

Keywords: deformation quantization; quantum gravity; Finsler and Lagrange geometry; almostK\" ahler geometry

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Zbl 1158.83010Vacaru, Sergiu I .P arametric nonholonom ic frame transforms and exact solutions in gravity. (English)


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[J] Int. J. Geom. Methods Mod. Phys. 4, No. 8, 1285-1334 (2007). ISSN 0219-8878

A geometric approach to constructing and classifying exact solutions in vacuum Einstein gravity isdeveloped and higher-dimensional theories of gravity by superposing parametric and anholonomicframe transformations. After outlining and comparing both the parametric and anholonomic framemethods of constructing exact solutions, a unified formalism for both formalisms is elaborated,where two alternative constructions are considered: either a class of solutions generated by theparametric method is deformed nonholonomically to the other ones and, inversely, the parametrictransform is applied to nonholonomic Einstein space-times. Finally, there are presented explicitexamples of how superpositions of both transformations can be applied in order to generate newclasses of solutions and how ``physically valuable configurations" can be selected. Someconstructions are performed for 5D space-times with torsion, but the bulk of them for 4D Einsteinspace-times with generic off diagonal metrics.[Horst-Heino von Borzeszkowski (Berlin) ]

MSC 2000:*83C15 Closed form solutions of equations in general relativity53B40 Finsler spaces and generalizations (local)

Keywords: exact solutions; Finsler geometry methods; nonlinear connections


Zbl 1153.81445Vacaru, Sergiu I .Deformation quantization of almost Kähler models and Lagrange-Finsler spaces. (English)[J] J. Math. Phys. 48, No. 12, 123509, 14 p. (2007). ISSN 0022-2488

Summary arXiv: Finsler and Lagrange spaces can be equivalently represented as almost Kählermanifolds enabled with a metric compatible canonical distinguished connection structuregeneralizing the Levi--Civita connection. The goal of this paper is to perform a naturalFedosov-type deformation quantization of such geometries [see {\it V. Karabegov} and {\it M.Schlichenmaier}, Lett. Math. Phys. 57, No. 2, 135--148 (2001; Zbl 1044.53061 )]. Allconstructions are canonically derived for regular Lagrangians and/or fundamental Finsler functionson tangent bundles. No review copy delivered.

MSC 2000:*53D55 Deformation quantization, star products53C60 Finsler spaces and generalizations (global)

Keywords: Fedosov-type deformation quantization; almost Kähler geometry; nonlinear connectionCitations: Zbl 1044.53061

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Zbl 1112.83010Vacaru, Sergiu I . ; Visinescu, MihaiNonholonomic Ricci flow s and running cosmological constant. I: 4D Taub-NUT metrics.(English)[J] Int. J. Mod. Phys. A 22, No. 6, 1135-1159 (2007). ISSN 0217-751X

Summary: In this work we construct and analyze exact solutions describing Ricci flows andnonholonomic deformations of four-dimensional (4D) Taub-NUT space--times. It is outlined a newgeometric technique of constructing Ricci flow solutions. Some conceptual issues on space--timesprovided with generic off-diagonal metrics and associated nonlinear connection structures areanalyzed. The limit from gravity/Ricci flow models with nontrivial torsion to configurations with the


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Levi-Civita connection is allowed in some specific physical circumstances by constraining the classof integral varieties for the Einstein and Ricci flow equations.

MSC 2000:*83C05 Einstein's equations83C15 Closed form solutions of equations in general relativity

Keywords: Ricci flows; exact solutions; Taub-NUT spaces


Zbl 1133.53307Vacaru, Sergiu I .Ricci flow s and solitonic pp-waves. (English)[J] Int. J. Mod. Phys. A 21, No. 23-24, 4899-4912 (2006). ISSN 0217-751X

Summary: We find exact solutions describing Ricci flows of four-dimensional pp-waves nonlinearlydeformed by two-/three-dimensional solitons. Such solutions are parametrized by five-dimensionalmetrics with generic off-diagonal terms and connections with nontrivial torsion which can berelated, for instance, to antisymmetric tensor sources in string gravity. There are definednontrivial limits to four-dimensional configurations and the Einstein gravity.

MSC 2000:*53C44 Geometric evolution equations (mean curvature flow)53C80 Appl. of global differential geometry to physics83C15 Closed form solutions of equations in general relativity83E30 String and superstring theories35Q51 Solitons

Keywords: Ricci flows; gravitational solitons; pp-waves


Zbl 1114.53062Vacar u, S. ; Stavrinos, P . ; Gaburov, E. ; Gonţa, D.( Anastasiei, Mihai )Clifford and Riemann-Fins ler structures in geo metric mechanics and gr avity. Selectedw orks. With a preface by Mihai Anastasiei. (English)[B] DGDS. Differential Geometry - Dynamical Systems. Monographs 7. Bucharest: Geometry of Balkan Press. xlix, 643~p. (2006).

The book contains a collection of works on Riemann-Cartan and metric-affine manifolds. Nonlinearconnection structures, generalized Finsler-Lagrange and Cartan-Hamilton geometries and Cliffordstructures are the main emphasis. Modelling locally anisotropic and/or noncommutative structuresin gravity and geometric mechanics are explicitely discussed. Einstein gravity theories and extradimension gravity, string models of gravity, black holes, anholonomic frame methods are examples

for the applications of the developed mathematical frame work. The overwhelming manifold of results will appeal to people interested in noncommutative and quantum developments of Finsler-Lagrange-Hamilton geometrics and nonholonomic structures in gravity and string theory.[Johannes Viktor Feitzinger (Bochum) ]

MSC 2000:*53C60 Finsler spaces and generalizations (global)53C80 Appl. of global differential geometry to physics83C20 Classes of solutions of equations in general relativity83C57 Black holes83E30 String and superstring theories70G45 Differential-geometric methods


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58A50 Supermanifolds, etc. (global analysis)81R60 Noncommutative geometry81T30 String and superstring theories

Keywords: Clifford structures; Riemann-Finsler structures; Einstein gravity; geometric mechanics;Black Holes; string theory

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Zbl 1112.53068Vacaru, Sergiu I .Clifford-Finsler algebroids and nonholonomic Einstein-Dirac structures. (English)[J] J. Math. Phys. 47, No. 9, 093504, 20 p. (2006). ISSN 0022-2488

Summary: We propose a new framework for constructing geometric and physical models on anonholonomic manifold provided both with Clifford-Lie algebroid symmetry and nonlinearconnection structure. Explicit parametrizations of generic off-diagonal metrics and linear andnonlinear connections define different types of Finsler, Lagrange, and/or Riemann-Cartan spaces.A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates thetheory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrabledistributions defining the nonlinear connection. In this work, we elaborate the algebroid spinor

differential geometry and formulate the (scalar, Proca, graviton, spinor, and gauge) field equationson Lie algebroids. The paper communicates new developments in geometrical formulation of physical theories and this approach is grounded on a number of previous examples when exactsolutions with generic off-diagonal metrics and generalized symmetries in modern gravity definenonholonomic spacetime manifolds with uncompactified extra dimensions.

MSC 2000:*53D99 Symplectic geometry, contact geometry58H05 Pseudogroups on manifolds


Zbl 1067.83011Vacaru, Sergiu I .Exact solution s w ith noncommutative symmetries in Einstein and gauge gravity. (English)[J] J. Math. Phys. 46, No. 4, 042503, 47 p. (2005). ISSN 0022-2488

Summary: We present new classes of exact solutions with noncommutative symmetriesconstructed in vacuum Einstein gravity (in general, with nonzero cosmological constant),five-dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated byanholonomic frame transforms and parametrized by generic off-diagonal metrics. For certainparticular cases, the new classes of metrics have explicit limits with Killing symmetries but, ingeneral, they may be characterized by certain anholonomic noncommutative matrix geometries.We argue that different classes of noncommutative symmetries can be induced by exact solutions

of the field equations in commutative gravity modeled by a corresponding moving real andcomplex frame geometry. We analyze two classes of black ellipsoid solutions (in the vacuum caseand with cosmological constant) in four-dimensional gravity and construct the analytic extensionsof metrics for certain classes of associated frames with complex valued coefficients. The third classof solutions describes 5D wormholes which can be extended to complex metrics in complex gravitymodels defined by noncommutative geometric structures. The anholonomic noncommutativesymmetries of such objects are analyzed. We also present a descriptive account how the Einsteingravity can be related to gauge models of gravity and their noncommutative extensions anddiscuss such constructions in relation to the Seiberg--Witten map for the gauge gravity. Finally,we consider a formalism of vielbeins deformations subjected to noncommutative symmetries inorder to generate solutions for noncommutative gravity models with Moyal (star) product.


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MSC 2000:*83C65 Methods of noncommutative geometry


Zbl 1067.53075Etayo, Fernando ; Santamaria, Rafael ; Vacaru, Sergiu I .Lagrange-Fedosov nonholonomic manifolds. (English)[J] J. Math. Phys. 46, No. 3, 032901, 17 p. (2005). ISSN 0022-2488

Summary: We outline a unified approach to geometrization of Lagrange mechanics, Finslergeometry and geometric methods of constructing exact solutions with generic off-diagonal termsand nonholonomic variables in gravity theories. Such geometries with induced almost symplecticstructure are modeled on nonholonomic manifolds provided with nonintegrable distributionsdefining nonlinear connections. We introduce the concept of Lagrange-Fedosov spaces andFedosov nonholonomic manifolds provided with almost symplectic connections adapted to thenonlinear connection structure. We investigate the main properties of generalized Fedosovnonholonomic manifolds and analyze exact solutions defining almost symplectic Einstein spaces.

MSC 2000:*53D55 Deformation quantization, star products81S10 Geometric quantization, symplectic methods


Zbl 1073.53034Vacaru, Sergiu I . ; Vicol, Nadejda A .Generalized Finsler superspaces. (English)[A] Tsagas, Grigorios (ed.), Proceedings of the conference on applied differential geometry andgeneral relativity and the workshop on global analysis, differential geometry and Lie algebras,Thessaloniki, Greece, 2002. Bucharest: Geometry Balkan Press; Bucharest: Balkan Society of Geometers. BSG Proceedings 11, 197-229 (2004). ISBN 973-8381-08-8/pbk

This paper applies general Finsler methods (generalized Lagrangian) to spaces withsupersymmetry. Various geometric structures such as nonlinear connections, torsions, andcurvatures are computed. Locally anisotropic spaces are studied and conditions for equivalence of supergravity models are developed.[Ralph G. Beil (Marshall) ]

MSC 2000:*53B40 Finsler spaces and generalizations (local)53B50 Appl. of local differential geometry to physics

Keywords: Finsler geometry; supergravity; local anisotropy; nonlinear connection


Zbl 1073.53033Tsagas, Grigorios ; Vacaru, Sergiu I .Nonlinear connections and isotopic Clifford structures. (English)[A] Tsagas, Grigorios (ed.), Proceedings of the conference on applied differential geometry andgeneral relativity and the workshop on global analysis, differential geometry and Lie algebras,Thessaloniki, Greece, 2002. Bucharest: Geometry Balkan Press; Bucharest: Balkan Society of Geometers. BSG Proceedings 11, 153-196 (2004). ISBN 973-8381-08-8/pbk

This is a general application of Finsler geometric methods to the isogeometry of Santilli. The ideasof Miron and Anastasiei, for example, using the nonlinear connection, appear to be compatible


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with isotopic products. This includes structures with Clifford fibration. It should be investigated,however, if isogeometry is not equivalent to generalized Lagrangian geometry with nonholonomicconnection.[Ralph G. Beil (Marshall) ]

MSC 2000:*53B40 Finsler spaces and generalizations (local)53B50 Appl. of local differential geometry to physics

Keywords: isogeometry; Clifford algebra; nonlinear connection; tangent bundlesPDF MathML XML ASCII DVI PS BibTeX Online Ordering

Zbl 1071.53043Vacaru, Sergiu I . ; Vicol, Nadejda A .Nonlinear connections and spinor geometry. (English)[J] Int. J. Math. Math. Sci. 2004, No. 21-24, 1189-1237 (2004). ISSN 0161-1712; ISSN1687-0425

Summary: We present an introduction to the geometry of higher-order vector and covectorbundles (including higher-order generalizations of the Finsler geometry and Kaluza-Klein gravity)

and review the basic results on Clifford and spinor structures on spaces with generic localanisotropy modeled by anholonomic frames with associated nonlinear connection structures. Weemphasize strong arguments for application of Finsler-like geometries in modern string andgravity theory, noncommutative geometry and noncommutative field theory, and gravity.

MSC 2000:*53C60 Finsler spaces and generalizations (global)15A66 Clifford algebras83C60 Spinor and twistor methods in general retativity53C27 Spin and Spin$^c$ geometry

Keywords: anholonomic frames; nonlinear connection; Finsler-like geometries; noncommutativegeometry


Zbl 1079.83531Vacaru, Sergiu I .Horizons and geodesics of black ellipsoids. (English)[J] Int. J. Mod. Phys. D 12, No. 3, 479-494 (2003). ISSN 0218-2718

Summary: We analyze the horizon and geodesic structure of a class of 4D off-diagonal metricswith deformed spherical symmetries, which are exact solutions of the vacuum Einstein equationswith anholonomic variables. The maximal analytic extension of the ellipsoid type metrics areconstructed and the Penrose diagrams are analyzed with respect to the adapted frames. We prove

that for small deformations (small eccentricities) there are such metrics that the geodesicbehaviour is similar to the Schwarzschild one. We conclude that some vacuum static andstationary ellipsoid configurations may describe black ellipsoid objects.



Zbl 1079.83530Vacaru, Sergiu I .


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Perturbations and stability of black ellipsoids. (English)[J] Int. J. Mod. Phys. D 12, No. 3, 461-478 (2003). ISSN 0218-2718

Summary: We study the perturbations of two classes of static black ellipsoid solutions of four-dimensional vacuum Einstein equations. Such solutions are described by generic off-diagonalmetrics which are generated by anholonomic transforms of diagonal metrics. The analysis isperformed in the approximation of small eccentricity deformations of the Schwarzschild solution.We conclude that such anisotropic black hole objects may be stable with respect to theperturbations parametrized by the Schrödinger equations in the framework of the one-dimensionalinverse scattering theory.



Zbl 1026.83049Dehnen, Heinz ; Vacaru, Sergiu I .Non linear connections and nearly autoparallel maps in general relativity. (English)[J] Gen. Relativ. Gravitation 35, No.5, 807-850 (2003). ISSN 0001-7701; ISSN 1572-9532

Summary: We apply the method of moving anholonomic frames with associated nonlinearconnections to the (pseudo) Riemannian space geometry and examine the conditions when locallyanisotropic structures (Finsler like and more general ones) could be modeled in the generalrelativity theory and/or Einstein-Cartan-Weyl extensions [{\it S. L. Vacaru} and {\it H. Dehnen},Gen. Relativ. Gravitation 35, 209-250 (2003; Zbl 1016.83034 )]. New classes of solutions of theEinstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps generalizing the conformal transforms and formulate the Einsteingravity theory on na-backgrounds provided with a set of na-map invariant conditions and localconservation laws. There are illustrated some examples when vacuum Einstein fields aregenerated by Finsler like metrics and chains of na-maps.

MSC 2000:

*83D05 Relativistic gravitational theories other than Einstein's83C10 Equations of motion53B40 Finsler spaces and generalizations (local)

Keywords: Anholonomic frame; Finsler metric; Einstein-Cartan-Weyl theoryCitations: Zbl 1016.83034


Zbl 1016.83034Vacaru, Sergiu I . ; Dehnen, HeinzLoc ally anisotropic structur es and nonlinear connections in Einstein and gauge gravity.

(English)[J] Gen. Relativ. Gravitation 35, No.2, 209-250 (2003). ISSN 0001-7701; ISSN 1572-9532

Summary: We analyze locally anisotropic configurations modeled by anholonomic frames withassociated nonlinear connections in general relativity, affine-Poincaré and/or de Sitter gaugegravity and Kaluza-Klein theories. A suitable geometrical formalism for theories with higher orderanisotropies and non compactified extra dimensions is introduced. We give a mostlyself-containing review of some aspects of gauge models of gravity and discuss their anholonomicgeneralizations and the conditions of equivalence with the Einstein gravity in arbitrarydimensions. New classes of cosmological solutions describing Friedmann-Robertson-Walker likeuniverses with resolution ellipsoid or torus symmetry.


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MSC 2000:*83D05 Relativistic gravitational theories other than Einstein's83E15 Higher-dimensional field theories

Keywords: anholonomic frames; Einstein gravity; gauge gravityCited in: Zbl 1026.83049


Zbl 1059.83034Vacaru, Sergiu I . ; Singleton, D.Ellipsoidal, cylindrical, bipolar and toroidal w ormholes in 5D gravity. (English)[J] J. Math. Phys. 43, No. 5, 2486-2504 (2002). ISSN 0022-2488

Summary: In this article we construct and analyze new classes of wormhole and flux tubelikesolutions for the 5D vacuum Einstein equations. These 5D solutions possess generic localanisotropy which gives rise to a gravitational running or scaling of the Kaluza-Klein ``electric"and "magnetic" charges of these solutions. It is also shown that it is possible to self-consistentlyconstruct these anisotropic solutions with various rotational 3D hypersurface geometries (i.e.,ellipsoidal, cylindrical, bipolar and toroidal). The local anisotropy of these solutions is handledusing the technique of anholonomic frames with their associated nonlinear connection structures[{\it S. I. Vacaru}, Ann. Phys. 256, No. 1, 39--61 (1997; Zbl 0956.83049 ); Nucl. Phys., B 494,No. 3, 590--656 (1997; Zbl 0934.81032 ); J. Math. Phys. 37, No. 1, 508--523 (1996; Zbl0870.53054 ); J. High Energy Phys. 1998, No. 9, Paper No. 11, 50 p. (1998; Zbl 0951.83031 );Phys. Lett., B 498, No. 1-2, 74--82 (2001; Zbl 0972.83047 )]. Through the use of the anholonomicframes the metrics are diagonalized, in contrast to holonomic coordinate frames where the metricswould have off-diagonal components. In the local isotropic limit these solutions are shown to beequivalent to spherically symmetric 5D wormhole and flux tube solutions.

MSC 2000:*83E05 Geometrodynamics83E15 Higher-dimensional field theories

Citations: Zbl 0956.83049 ; Zbl 0870.53054 ; Zbl 0951.83031 ; Zbl 0972.83047 ; Zbl 0934.81032PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

Zbl 1007.83032Vacaru, Sergiu I . ; Singleton, D.W arped, anisotropic w ormhole/ soliton configurations in vacuum 5D gravity. (English)[J] Classical Quantum Gravity 19, No.11, 2793-2811 (2002). ISSN 0264-9381; ISSN 1361-6382

Summary: We apply the anholonomic frames method developed in previous work to construct andstudy anisotropic vacuum field configurations in 5D gravity. Starting with an off-diagonal 5Dmetric, parametrized in terms of several ansatz functions, we show that using anholonomic framesgreatly simplifies the resulting Einstein field equations. These simplified equations contain aninteresting freedom in that one can choose one of the ansatz functions and then determine theremaining ansatz functions in terms of this choice. As examples we take one of the ansatzfunctions to be a solitonic solution of either the Kadomtsev-Petviashvili equation or thesine-Gordon equation. There are several interesting physical consequences of these solutions.First, a certain subclass of the solutions discussed in this paper has an exponential warp factorsimilar to that of the Randall-Sundrum model. However, the warp factor depends on more than

just the fifth coordinate. In addition the warp factor arises from anisotropic vacuum solutionsrather than from any explicit matter. Second, the solitonic character of these solutions might allowthem to be interpreted either as gravitational models for particles (i.e. analogous to the 't Hooft-Polyakov monopole, but in the context of gravity), or as nonlinear, anisotropic gravitationalwaves.


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MSC 2000:*83E15 Higher-dimensional field theories53Z05 Appl. of differential geometry to physics83C35 Gravitational waves

Keywords: anisotropic vacuum field configurations; 5D gravity; sine-Gordon equation; warpfactor; anisotropic gravitational waves


Zbl 1005.83029Vacaru, Sergiu I . ; Singleton, D.W arped solitonic defor mations and propagation of black holes in 5D vacuum gravity.(English)[J] Classical Quantum Gravity 19, No.14, 3583-3601 (2002). ISSN 0264-9381; ISSN 1361-6382

This paper is devoted to the study of warped solitonic deformations and propagation of black holesin 5D vacuum gravity. The authors present the theoretical framework of the anholonomic framesmethod, introduce the 5D metric ansatz form and write down the corresponding vacuum Einsteinequations, discuss the sine-Gordon soliton, gravitational soliton, 4D Schwarzschild solutionembedded in the 5D spacetime, solitonic black hole solutions which move in the bulk 5D spacetime

and show that these solitonic solutions may either violate or preserve local Lorentz invariance,and these solutions may point to Lorentz violation effects in Yang-Mills and electrodynamictheories induced from 5D vacuum gravity.

MSC 2000:*83E15 Higher-dimensional field theories83C57 Black holes

Keywords: black holes; 5D vacuum gravity; solitonic deformations; Yang-Mills theories


Zbl 0989.83040Vacaru, Sergiu I . ; Tintareanu-Mircea, OvidiuA nholono mic frames, generalized Killing equations, and anisotropic Taub-NUT spin ningspaces. (English)[J] Nucl. Phys., B 626, No.1-2, 239-264 (2002). ISSN 0550-3213

Summary: By using anholonomic frames in (pseudo)-Riemannian spaces we define anisotropicextensions of Euclidean Taub-NUT spaces. With respect to coordinate frames such spaces aredescribed by off-diagonal metrics which could be diagonalized by corresponding anholonomictransforms. We define the conditions when the 5D vacuum Einstein equations have as solutionsanisotropic Taub-NUT spaces. The generalized Killing equations for the configuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed. Simple solutions of thehomogeneous part of these equations are expressed in terms of some anisotropically modifiedKilling-Yano tensors. The general results are applied to the case of the four-dimensional locallyanisotropic Taub-NUT manifold with Euclidean signature. We emphasize that all constructions arefor (pseudo)-Riemannian spaces defined by vacuum solutions, with generic anisotropy, of 5DEinstein equations, the solutions being generated by applying the moving frame method.

MSC 2000:*83E05 Geometrodynamics

Keywords: pseudo-Riemannian spaces; off-diagonal metrics; 5D vacuum Einstein equations



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Zbl 1062.82047Vacaru, Sergiu I .Loc ally anisotropic kinetic processes and thermodynamics in curved spaces. (English)[J] Ann. Phys. 290, No. 2, 83-123 (2001). ISSN 0003-4916

Summary: The kinetic theory is formulated with respect to anholonomic frames of reference oncurved spacetimes. By using the concept of nonlinear connection we develop an approach tomodelling locally anisotropic kinetic processes and, in corresponding limits, the relativisticnonequilibrium thermodynamics with local anisotropy. This leads to a unified formulation of thekinetic equations on (pseudo) Riemannian spaces and in various higher dimensional models of Kaluza-Klein type and/or generalized Lagrange and Finsler spaces. The transition rate consideredfor the locally anisotropic transport equations is related to the differential cross section andspacetime parameters of anisotropy. The equations of states for pressure and energy in locallyanisotropic thermodynamics are derived. The obtained general expressions for heat conductivity,shear, and volume viscosity coefficients are applied to determine the transport coefficients of cosmic fluids in spacetimes with generic local anisotropy. We emphasize that such local anisotropicstructures are induced also in general relativity if we are modelling physical processes with respectto frames with mixed sets of holonomic and anholonomic basis vectors which naturally admits anassociated nonlinear connection structure.

MSC 2000:

*82C40 Kinetic theory of gases53B40 Finsler spaces and generalizations (local)53C80 Appl. of global differential geometry to physics83C99 General relativity


Zbl 0987.83038Vacaru, Sergiu I . ; P opa, Florian CatalinDirac spinor w aves and solitons in anisotropic Taub-NUT spaces. (English)[J] Classical Quantum Gravity 18, No.22, 4921-4938 (2001). ISSN 0264-9381; ISSN 1361-6382

Summary: We apply a new general method of anholonomic frames with associated nonlinearconnection structure to construct new classes of exact solutions of Einstein-Dirac equations infive-dimensional (5D) gravity. Such solutions are parametrized by off-diagonal metrics incoordinate (holonomic) bases or equivalently, by diagonal metrics given with respect to someanholonomic frames (pentads or fuenfbeins satisfying corresponding constraint relations). Weconsider two possibilities of generalization of the Taub-NUT metric in order to obtain vacuumsolutions of 5D Einstein equations with effective renormalization of constants (by higherdimension anholonomic gravitational interactions) having distinguished anisotropies on an angularparameter or on an extra-dimensional coordinate. The constructions are extended to solutionsdescribing self-consistent propagations of 3D Dirac wave packets in 5D anisotropic Taub-NUTspacetimes. We show that such anisotropic configurations of spinor matter can induce gravitational3D solitons which are solutions of Kadomtsev-Petviashvili or sine-Gordon equations.

MSC 2000:*83C60 Spinor and twistor methods in general retativity83C57 Black holes83E15 Higher-dimensional field theories

Keywords: spinor waves; Einstein-Dirac equations; five-dimensional (5D) gravity; Taub-NUTspacetimes


Zbl 0972.83021


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Vacaru, S.I. ; Singleton, D. ; Boţan, V.A. ; Doţenco, D.A.Locally anisotropic w ormholes and flux tubes in 5D gravity. (English)[J] Phys. Lett., B 519, No.3-4, 249-259 (2001). ISSN 0370-2693

Summary: In this letter we examine a class of wormhole and flux tube like solutions to 5D vacuumEinstein equations. These solutions possess generic local anisotropy, and their local isotropic limitis shown to be conformally equivalent to the spherically symmetric 5D solutions by {\it V.Dzhunushaliev} and {\it D. Singleton} [Phys. Rev. D (3) 59064018, No. 6, 064018 (1999)]. Theanisotropic solutions investigated here have two physically distinct signatures: first, they can giverise to angular-dependent, anisotropic ``electromagnetic'' interactions. Second, they can result ina gravitational running of the ``electric'' and ``magnetic'' charges of the solutions. Thisgravitational running of the electromagnetic charges is linear rather than logarithmic, and couldthus serve as an indirect signal for the presence of higher dimensions. The local anisotropy of these solutions is modeled using the technique of anholonomic frames with respect to which themetrics are diagonalized. If holonomic coordinate frames were used then such metrics would haveoff-diagonal components.

MSC 2000:*83C15 Closed form solutions of equations in general relativity83E15 Higher-dimensional field theories83E05 Geometrodynamics

Keywords: 5D vacuum Einstein equation solutions; gravitational electromagnetic charge running;metric diagonalization


Zbl 0972.83047Vacaru, S.I.Gauge and Einstein gravity from non-abelian gauge models on noncommu tative spaces.(English)[J] Phys. Lett., B 498, No.1-2, 74-82 (2001). ISSN 0370-2693

Summary: Following the formalism of enveloping algebras and star product calculus we formulateand analyze a model of gauge gravity on noncommutative spaces and examine the conditions of its equivalence to the general relativity theory. The corresponding Seiberg-Witten maps areestablished which allow the definition of respective dynamics for a finite number of gravitationalgauge field components on noncommutative spaces.

MSC 2000:*83C65 Methods of noncommutative geometry81T13 Gauge theories81T75 Noncommutative geometry methods83C45 Quantization of the gravitational field81S10 Geometric quantization, symplectic methods

Keywords: Seiberg-Witten maps; enveloping algebras; star product calculusCited in: Zbl 1059.83034


Zbl 1006.53066Vacaru, Sergiu I .Nearly autoparallel maps, tensor integral and conservation law s on locally anisotro picspaces. (English)[A] Gill, Tepper (ed.) et al., Fundamental open problems in science at the end of the millennium.Proceedings of the international workshop, Beijing, China, August 1997. Vols. I-III. Palm Harbor,


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FL: Hadronic Press. 67-103 (1999). ISBN 1-57485-029-6/pbk

The author proposes a certain generalization of conformal transformations which he calls `nearlyautoparallel maps''. In the resulting theory the proofs of most of the results are, to use theauthor's own description, mechanical and tedious. The theory is supposed to be applicable to theformulation of conservation laws for locally anisotropic gravity, though the theory has not yetreached a point where it can present an experimentally verifiable result.[Chandra Shekhar Sharma (London) ]

MSC 2000:*53C80 Appl. of global differential geometry to physics83C40 Groups of motions, etc.53C07 Special connections and metrics on vector bundles83E15 Higher-dimensional field theories

Keywords: autoparallel maps; conformal transformation; anisotropic space


Zbl 0977.83130Vacaru, Sergiu I .

Exact solutions in locally anisotropic gravity and strings. (English)[A] Rembieliński, Jakub (ed.), Particles, fields, and gravitation. Papers from the c onferencededicated to the memory of Ryszard Raczka, Łódz, Poland, April 15-19, 1998. Woodbury , NY:American Institute of Physics. AIP Conf. Proc. 453, 528-537 (1998). ISBN 1-56396-837-1

Summary: In this report we outline some basic results on generalized Finsler-Kaluza-Klein gravityand locally anisotropic strings. There are investigated exact three-dimensional solutions for locallyanisotropic Friedmann-Robertson-Walker universes and string black holes with generic anisotropy.

MSC 2000:*83F05 Relativistic cosmology53B40 Finsler spaces and generalizations (local)83E15 Higher-dimensional field theories83C57 Black holes83C80 Analogues in lower dimensions83D05 Relativistic gravitational theories other than Einstein's83E30 String and superstring theories

Keywords: gravity; strings; Finsler-Kaluza-Klein gravity; Friedmann-Robertson-Walker universes;string black holes


Zbl 0954.53049Vacaru, Sergiu I on

Interactions, strings and i sotopies in higher order anisotropic superspaces. (English)[B] Hadronic Press Monographs in Mathematics. Palm Harbor, FL: Hadronic Press. 448 p. \sterling45.00; \$ 85.00 (1998). ISBN 1-57485-032-6/pbk

In some sense, this is a book of everything. The author's ambitious formalism is applicable to mostof the branches of mathematical physics, including gauge theories, strings, spinors, superstrings,conservation laws and diffusion processes. The mathematical tools for this aggregation are foundin Finsler geometry and its generalization to Lagrange spaces, prolongations of this geometry tohigher order fiber bundles, and the further extension to stochastic processes. The speculativeisogeometry of Santilli is also encompassed. This difficult task is accomplished quite competently.The only problem is that the resulting system is so complicated that it takes a concerted effort toget through to the physics. In a forest so dense it is difficult to find the right tree. The book has an


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adequate bibliography, but suffers from the lack of an index.[R.G.Beil (Marshall) ]

MSC 2000:*53Z05 Appl. of differential geometry to physics81-01 Textbooks (quantum theory)83-01 Textbooks (relativity)53B40 Finsler spaces and generalizations (local)53B50 Appl. of local differential geometry to physics

81Q60 Supersymmetric quantum mechanics81T30 String and superstring theories83C60 Spinor and twistor methods in general retativity83E15 Higher-dimensional field theories83E30 String and superstring theories

Keywords: Finsler geometry; strings; spinors; superstrings; stochastic analysis; Kaluza-Kleintheory; Lagrangian geometry


Zbl 0951.83031

Vacaru, Sergiu I .Spinors and field interactions in higher order anisotropi c spaces. (English)[J] J. High Energy Phys. 1998, No.9, Paper No.11, 50 p., electronic only (1998). ISSN 1029-8479

Summary: We formulate the theory of field interactions with higher order anisotropy. Theconcepts of higher order anisotropic space and locally anisotropic space (in brief, ha-space andla-space) are introduced as general ones for various types of higher order extensions of Lagrangeand Finsler geometry and higher dimension (Kaluza-Klein type) spaces. The spinors on ha-spacesare defined in the framework of the geometry of Clifford bundles provided with compatiblenonlinear and distinguished connections and metric structures ($d$-connection and $d$-metric).The spinor differential geometry of ha-spaces is constructed. There are discussed some relatedissues connected with the physical aspects of higher order anisotropic interactions forgravitational, gauge, spinor, Dirac spinor and Proca fields. Motion equations in higher ordergeneralizations of Finsler spaces, of the mentioned type of fields, are defined by using bundles of linear and affine frames locally adapted to the nonlinear connection structure.

MSC 2000:*83E15 Higher-dimensional field theories53C15 Geometric structures on manifolds53C60 Finsler spaces and generalizations (global)53C80 Appl. of global differential geometry to physics81T20 Quantum field theory on curved space backgrounds

Keywords: ha-space; locally anisotropic space; Clifford bundle; Finsler spaces; la-spaceCited in: Zbl 1059.83034

PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article JournalJournal

Zbl 0956.83049Vacaru, Sergiu I .Locally anisotropic gravity and strings. (English)[J] Ann. Phys. 256, No.1, 39-61 (1997). ISSN 0003-4916

Differential geometric methods play an important role in description and investigation of problemsof gravity and strings. The paper under review consists of 6 Sections. In Section 1 the author


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briefly discusses connections between gravity and strings and locally anisotropic (la) spaces andgives the plan of the paper. Section 2 contains a geometric background of vector bundles providedwith nonlinear and distinguished connections and metric structure. In Section 3 the authorpresents a generalization of some necessary results on the non-linear $\sigma-$model and stringpropagation to the case of la-backgrounds. The aim of Section 4 is to study the problem of regularization and quantum ambiguities in $\beta-$functions of the renormalization group and topresent the results on one- and two-loop calculi for the la-$\sigma-$model. In Section 5 theauthor investigates duality of la-$\sigma-$model. The paper ends with conclusions, preceded by asummary of the author's results (Section 6).[Nikolaj M.Glazunov (Ky{\" i}v) ]

MSC 2000:*83E30 String and superstring theories81T30 String and superstring theories81T17 Renormalization group methods83E15 Higher-dimensional field theories53B50 Appl. of local differential geometry to physics

Keywords: locally anisotropic (LA) space; gravity; strings; N-connection; regularization; duality of LA sigma-modelCited in: Zbl 1059.83034


Zbl 0934.81032Vacaru, Sergiu I .Superstrings in higher order extensions of Finsler supers paces. (English)[J] Nucl. Phys., B 494, No.3, 590-656 (1997). ISSN 0550-3213

Summary: The work proposes a general background of the theory of field interactions and stringsin spaces with higher order anisotropy. Our approach proceeds by developing the concept of higherorder anisotropic superspace which unifies the logical and mathematical aspects of modernKaluza-Klein theories and generalized Lagrange and Finsler geometry and leads to modelling of

physical processes on higher order fiber bundles provided with nonlinear and distinguishedconnections and metric structures. The view adopted here is that a general field theory shouldincorporate all possible anisotropic and stochastic manifestations of classical and quantuminteractions and, in consequence, a corresponding modification of basic principles andmathematical methods in formulation of physical theories.\par The presentation is divided into twoparts. The first five sections cover the higher order anisotropic superspaces. We focus on thegeometry distinguished by non-linear connection vector superbundles, consider differentsupersymmetric extensions of Finsler and Lagrange spaces and analyze the structure of basicgeometric objects on such superspaces. The remaining five sections are devoted to the theory of higher order anisotropic superstrings. In the framework of supersymmetric nonlinear sigmamodels in Finsler extended backgrounds we prove that the low-energy dynamics of such stringscontains equations of motion for locally anisotropic field interactions. Our work is to be compared

with important previous variants of extensions of Finsler geometry and gravity.MSC 2000:

*81T30 String and superstring theories53C60 Finsler spaces and generalizations (global)58A50 Supermanifolds, etc. (global analysis)

Keywords: Kaluza-Klein theories; Lagrange geometry; Finsler geometry; field theory;supersymmetric Lagrange spaces; equations of motionCited in: Zbl 1059.83034



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Zbl 0914.53039Vacaru, Sergiu I .Studies on Santilli's loc ally anisotropi c and inhomogeneous isog eometries. I: Iso bundlesand generalized isofinsler gravity. (English)[J] Algebras Groups Geom. 14, No.3, 211-257 (1997). ISSN 0741-9937

The aim of the present paper is the formulation of the theory of the inhomogeneous and locallyanisotropic isofield interactions. This contains a synthesis of Santilli isotheory and the approach onmodeling locally anisotropic geometries and physical models on bundle spaces provided withnonlinear connection and distinguished connection and metric structures. The isotopic variants of generalized Lagrange and Finsler geometry are analyzed. Basic geometric constructions such asnonlinear isoconnections in vector isobundles, the isotopic curvatures and torsions of distinguishedisoconnections and their structure equations and invariant values are defined. A model of locallyanisotropic and inhomogeneous gravitational isotheory is constructed. \par The paper contains thefollowing headings. Isotopies of the unit and isospaces, Isocontinuity and isotopology,Isodifferential and isointegral calculus, Santil li's isoriemannian isospaces, Lie-Santilli isoalgebrasand isogroups, Fiber isobundles, Isobianchi and isoricci identities, Structure equations of a$d$-isoconnection, Notions of generalized isolagrange and isofinsler spaces and the isotopic almostHermitian model of the GL-space.[G.Tsagas (Thessaloniki) ]

MSC 2000:*53C99 Global differential geometry

Keywords: isolagrance space; isofinsler space; isotopic almost Hermitian model of GL-space;inhomogeneous and locally anisotropic isofield interactions


Zbl 0905.35101Vacaru, Sergiu I .Locally anisotropic stochastic processes in fiber bundles. (English)[A] Tsagas, Grigorios (ed.), Proceedings of the workshop on global analysis, differential geometryand Lie algebras, Aristotle University of Thessaloniki, Greece, December 16--18, 1995. Bucharest:Geometry Balkan Press. BSG Proc. 1, 123-140 (1997). ISBN 973-97454-1-5/pbk

The geometrical basis of the theory of diffusion processes on spaces with local anisotropy isconsidered. Such spaces are modeled as vector bundles on space-times provided with nonlinearand distinguished connections ($N$- and $d$-connections, respectively) and metric structures.\par Applying those considerations to tangent bundles, the author formulates the theory of stochastic differential equations on generalized Lagrange spaces, which contain as particular casesLagrange and Finsler spaces. The author also gives some remarks on possible extensions of theresults presented in the paper. The biblography contains 43 items. \par It is worth to point out thepaper [{\it P. L. Antonelli} and {\it T. J. Zastawniak}, Nonlinear World 1, No. 2, 149-171 (1994;Zbl 0799.60057 )] on the theory of diffusion on Finsler manifolds with applications to biology.[W.Kotarski (Sosnowiec) ]

MSC 2000:*35R60 PDE with randomness53B15 Other connections60H15 Stochastic partial differential equations

Keywords: vector bundles; connections; tangent bundles; generalized Lagrange spaces; FinslerspacesCitations: Zbl 0799.60057



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Zbl 1198.35293Văcaru, Sergiu I.Stochastic differential equations on spaces w ith local anisotropy. (English)[J] Bul. Acad. Ştiinţe Repub. Mold., Fiz. Teh. 1996, No. 2(20), 154-174 (1996). ISSN 0236-3097

MSC 2000:*35R60 PDE with randomness53B15 Other connections60H15 Stochastic partial differential equations


Zbl 1198.53087Văcaru, Sergiu I.Y ang--Mills fields on spaces w ith local anisotropy. (English)[J] Bul. Acad. Ştiinţe Repub. Mold., Fiz. Teh. 1996, No. 1(19), 41-44 (1996). ISSN 0236-3097

MSC 2000:*53C80 Appl. of global differential geometry to physics83E15 Higher-dimensional field theories53C15 Geometric structures on manifolds


Zbl 1198.53086Văcaru, Sergiu I.Gauge like treatment of generalized Lagrange and Fins ler gravity. (English)[J] Bul. Acad. Ştiinţe Repub. Mold., Fiz. Teh. 1996, No. 1(19), 35-40 (1996). ISSN 0236-3097

MSC 2000:*53C80 Appl. of global differential geometry to physics83E15 Higher-dimensional field theories53C60 Finsler spaces and generalizations (global)


Zbl 0890.53068Vacaru, Sergiu I . ; Ostaf, Sergiu V.Tw istors and nearly autoparallel maps. (English)[J] Rep. Math. Phys. 37, No.1-3, 309-324 (1996). ISSN 0034-4877

As is well-known, the twistor equation on a non-conformally flat spacetime $V$ is not compatible.The authors idea in the present paper is to ``transport'' the twistor equation from the Minkowskispace to $V$ by means of a nearly geodesic map in the sense of {\it N. S. Sinyukov} [`Geodesicmappings of Riemannian spaces' (Nauka, Moscow) (1979; Zbl 0637.53020 )] (or by a chain of such

maps). In this way they obtain a spinor equation on $V$ whose solutions may be considered astwistors on $V$. The authors also give conditions under which the transformed twistor structuregenerates to a vacuum Einstein field.[ J.Davidov (Sofia) ]

MSC 2000:*53Z05 Appl. of differential geometry to physics83C60 Spinor and twistor methods in general retativity

Keywords: twistors; nearly geodesic mapsCitations: Zbl 0637.53020


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Zbl 0870.53054Văcaru, Sergiu I.Spinor structures and no nlinear connections in vector bundles, generalized Lagrange andFinsler spaces. (English)[J] J. Math. Phys. 37, No.1, 508-523 (1996). ISSN 0022-2488

Let $\xi= (E,p,M)$ be a vector bundle over a smooth manifold $M$. The kernel of the Jacobi map$Dp:TE\to TM$ provides the vertical distribution $u\to V_uE$, $u\in E$, on $E $. A distribution$u\to H_uE$ such that $(*)$ $T_u E=H_uE \oplus V_uE$ is called a nonlinear connection on $E$.The splitting $(*)$ produces a decomposition of all geometrical objects (tensors, connections etc.)on $E$ and their components are generically called $d$-objects [see {\it R. Miron} and {\it M.Anastasiei}, The geometry of Lagrange spaces: theory and applications (Fundamental Theories of Physics 59, Kluwer, Dordrecht) (1994; Zbl 0831.53001 )]. A metrical structure in the verticalbundle is called a $d$-metric. This can be extended to a metrical structure $G$ on $E$ and thereexists a linear connection $D$ on $E$ which is metrical and preserves the splitting $(*)$.\par Inthis paper a spinor formalism for $\xi$ endowed with a nonlinear connection, the metric $G$ andthe connection $D$ is proposed. Clifford $d$-algebras, a twisted Clifford $d$-group, and thegroups Pin and Spin are introduced. The spinor structure is defined in terms of principal bundles. Acomplex version is provided. The periodicity phenomena are analyzed.[M.Anastasiei (Iaşi) ]

MSC 2000:*53C60 Finsler spaces and generalizations (global)53C27 Spin and Spin$^c$ geometry

Keywords: Lagrange and Finsler spaces; spin structures; Clifford algebra; $d$-metric; nonlinearconnectionCitations: Zbl 0831.53001Cited in: Zbl 1059.83034


Zbl 0846.53014Gottlieb, I. ; Văcaru, Sergiu I.A . Moó r's tensorial integration in generalized Lagrange spaces. (English)[A] Antonelli, P. L. (ed.) et al., Lagrange and Finsler geometry: applications to physics and biology.Proceedings of a conference. Dordrecht: Kluwer Academic Publishers. Fundam. Theor. Phys. 76,209-216 (1996). ISBN 0-7923-3873-1/hbk

The problem of tensorial integration as the inverse operator to covariant derivation was proposedand studied by {\it A. Moór} [Acta Math. 86, 71-83 (1951; Zbl 0044.37301 ); Monatsh. Math. 70,134-148 (1966; Zbl 0139.15301 )] and others. Tensor integral methods have been used in solvingcertain difficulties in the formulation of conservation laws on curved spaces in the framework of general relativity [{\it I. Gottlieb}, Period. Mat. Hung. 8, 57-64 (1977; Zbl 0354.53009 )]. Thepresent paper is devoted to the generalization of the tensor integral theory to generalizedLagrange spaces.[M.Matsumoto (Kyoto) ]

MSC 2000:*53B40 Finsler spaces and generalizations (local)53A45 Vector and tensor analysis

Keywords: tensorial integration; generalized Lagrange spacesCitations: Zbl 0044.37301 ; Zbl 0139.15301 ; Zbl 0354.53009


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Zbl 0844.53047Văcaru, Sergiu I. ; Ostaf, S.Nearly autoparallel maps of Lagrange and Finsl er spaces. (English)[A] Antonelli, P. L. (ed.) et al., Lagrange and Finsler geometry: applications to physics and biology.Proceedings of a conference. Dordrecht: Kluwer Academic Publishers. Fundam. Theor. Phys. 76,241-253 (1996). ISBN 0-7923-3873-1/hbk

Let $L = (M, {\cal L})$ be a Lagrange space (i.e. such a generalization of the Finsler space, wherethe Lagrangian ${\cal L} : TM \to \bbfR$) ${\cal L} : (x,y) \mapsto {\cal L} (x,y)$ is notnecessarily homogeneous of degree 2 in $y$. The authors call a curve $\gamma (x (t), y(t))\subset L$ distinguished autoparallel (da-parallel) if $D_Y Y = \rho(t) Y$, where $Y(t) = (\dot x(t),\dot y(t))$ and $D$ is the canonical distinguished connection. A 2-dimensional distribution $E_2(\gamma)$ along $\gamma$ is called coplanar if the parallel translated of any vector $X_0 \inE_2 (\gamma (t_0))$ is contained in $E_2(\gamma)$. $\gamma$ is called distinguished nearlyautoparallel (dna-parallel) if there exists a coplanar $E_2(\gamma)$ containing $Y(t)$. In thepaper 1-1 mappings $L \to \underline {L}$ are studied and classified which take every da-parallelcurve of $L$ into a dna-parallel of $\underline{L}$. Also physical applications are considered.[L.Tamássy (Debrecen) ]

MSC 2000:*53C60 Finsler spaces and generalizations (global)

Keywords: generalized projective mappings of Lagrange spaces; Lagrange space; Finsler space


Zbl 1198.53085Văcaru, Sergiu I.Clifford structures and spinors on spaces w ith local anisotropy. (English)[J] Bul. Acad. Ştiinţe Repub. Mold., Fiz. Teh. 1995, No. 3(18), 53-62 (1995). ISSN 0236-3097

MSC 2000:*53C80 Appl. of global differential geometry to physics83E15 Higher-dimensional field theories53C15 Geometric structures on manifolds53C60 Finsler spaces and generalizations (global)81T20 Quantum field theory on curved space backgrounds


Zbl 0842.53020Văcaru, Sergiu I. ; Goncharenko, YuriiY ang-Mills fields and gauge gravity on generalized Lagrange and Finsler spaces. (English)

[J] Int. J. Theor. Phys. 34, No.9, 1955-1980 (1995). ISSN 0020-7748; ISSN 1572-9575

In the framework of the theory of linear connections in vector bundles (with semisimple structuralgroups) on generalized Lagrange spaces, a geometrical approach to interactions of Yang-Millsfields on spaces with local anisotropy is formulated. The geometrical formalism is extended in amanner including theories with nonsemisimple groups which permit a unique fiber bundletreatment for both locally anisotropic Yang-Mills and gravitational interactions.\par One of themost important results of the paper is formulated as a theorem stating that almost HermitianLagrange gravity -- described in {\it R. Miron} and {\it M. Anastasiei} [The geometry of Lagrangespaces: theory and applications. Fundamental Theories of Physics, 59, Dordrecht: KluwerAcademic Publishers (1994; Zbl 0831.53001 )], is equivalent to a gaugelike theory in the bundle of affine adapted frames on generalized Lagrange spaces.


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[V.Balan (Bucureşti) ]

MSC 2000:*53C07 Special connections and metrics on vector bundles81T13 Gauge theories83C47 Quantum field theory on curved space-times53Z05 Appl. of differential geometry to physics

Keywords: Finsler spaces; generalized Lagrange geometry; Yang-Mills equations

Citations: Zbl 0831.53001PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

Zbl 0867.53022Văcaru, Sergiu I. ; Ostaf, S. ; Goncharenko, Yu. ; Doina, A.Nearly autoparallel maps of Lagrange spaces. (English)[J] Bul. Acad. Ştiinţe Repub. Mold., Fiz. Teh. 1994, No.3(15), 42-53 (1994). ISSN 0236-3097

Let $M_{\underline m}$ be an $n$-dimensional manifold and $TM_{\underline m}$ be itstangent bundle with local coordinates $x_{\underline m}= xî_{\underline m}$ and$(xî_{\underline m},y j_{\underline m})$, $i,j=1,2,\dots,n$, respectively. A Lagrangian on

$M_{\underline m}$ is a differentiable function $L_{\underline m}:TM_{\underline m}\to R$,given locally by $L_{\underline m}:(x,y)\to L_{\underline m}(x,y)$ such that the tensorialdistinguished field $g_{\underline m ij}(x,y){1\over 2}(\partial^2L_{\underline m}/\partialyî\partial y^j)$ is nondegenerate. A pair $(M_{\underline m},L_{\underline m})$, simplydenoted by $L$, is called a Lagrangian space of dimension $n$ with $L_{\underline m}$ and$g_{\underline m ij}$ as the fundamental function and metric tensor, respectively. A Finslerspace is a particular case of a Lagrange space $(M_{\underline m},L_{\underline m})$ when$L_{\underline m}= L^2_{\underline m}$ which is the Finsler metric on $M_{\underlinem}$.\par Let $L, L_{\underline 1},L_{\underline 2},\dots,L_{\underline n}$, where $\underlinek,\underline l ,\underline m,\underline n=1,2,3,\dots$ (underlined subscripts indicate differentspaces) be Lagrange spaces. Consider $1-1$ local mappings $f:L_{\underline m}\to L_{\underlinek}$, $m\ne k$ with deformation of nonlinear connections in general. A curve

$\gamma_{\underline m}$ on $L$ is a function $\gamma_{\underline m}:R\to L$, representedparametrically as $\gamma_{\underline m}= (x(\eta),y(\eta))$ with parameter $\eta$,$(\eta_1\le \eta\le\eta_2)$, and the tangent vector to the curve $\gamma_{\underline m}$ by$Y(\eta)= ({\partial x\over\partial\eta},{\partial y\over\partial\eta})\in TL_{\underline m}$.When there is defined a two-dimensional vector space $E_2\subset TL_{\underline m}(x)$ atevery point $x\in\gamma_{\underline m}$, we say that the curve $\gamma_{\underline m}$ isprovided with two-dimensional distribution $E_2$. A distribution $E_2(\gamma_{\underline m})$is said to be coplanar along $\gamma_{\underline m}$ if every vector $X_{m(0)}=X_m(x_0,y_0)\in E_2(\gamma_{\underline m})$, $(x_0,y_0)\in\gamma_{\underline m}$, iscontained in the same distribution after parallel transport along $\gamma_{\underline m}$, i.e.,$X_{\underline m}(X(\eta),Y(\eta))\subset E_2(\gamma_{\underline m})$, $\forall\eta\in(\eta_1,\eta_2)$.\par The curve $\gamma$ on the Lagrange space $L$ is called

distinguished auto(da-)parallel if the tangent vector field $Y$ to $\gamma$ satisfies theauto-parallel equations $D_YY= \rho(\eta)Y$, where $\rho(\eta)$ is a scalar function on $L$. Thecurve $\gamma$ is called distinguished nearly auto(dna-)parallel if there is defined atwo-dimensional coplanar distribution $E_2(\gamma)$ containing the tangent vector field $Y$ to$\gamma$. Moreover, let $L$ and $\underline L$ be two Lagrange spaces. The local $1-1$mappings $na:L\to\underline L$, which change every da-parallel curve on $L$ into a dna-parallelcurvature on $\underline L$, are called nearly auto(na-)parallel maps.\par In the present paper,having defined na-maps, the authors discuss deformation of fundamental function, metric,nonlinear and canonical connections under such a map and obtain the basic deformationequations. Taking clue from these, the authors classify all na-maps into four types yielding thecorresponding basic equations which are systems of first-order partial differential equations withalgebraic constraints. The authors give a new classification of Lagrange spaces based on chains of


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na-maps and thus introduce the concept of Lagrange spacetime category $LC_{st}$ with objectsbeing Lagrange spaces and morphisms being chains of na-maps of Lagrange spaces. In the end,the authors consider fundamental spacetime $V$, Lagrange space $L$ and a chain of na-mapsfrom $V$ to $L$ and then transport the Einstein equation from $V$ to $L$ so as to explore itunder na-backgrounds.[O.P.Singh (Aligarh) ]

MSC 2000:*53B40 Finsler spaces and generalizations (local)

53C60 Finsler spaces and generalizations (global)

Keywords: nearly autoparallel maps; classification of Lagrange spaces; Lagrange spacetime


Zbl 0837.53062Văcaru, Sergiu I. ; Goncharenko, Yu.A.On exactly solvable 4D quantum gravities. (English)[J] Bul. Acad. Ştiinţe Repub. Mold., Fiz. Teh. 1994, No.3(15), 53-58 (1994). ISSN 0236-3097

Authors' summary: ``New classes of nearly autoparallel maps and their superpositions, called

nearly conformal transforms and generalizing conformal rescalings, are introduced. We describebriefly how to obtain exactly solvable $4d$ gravitational models by using nearly conformal mapsdirected by a $2d$ quantum gravity''.[A.D.Osborne (Keele) ]

MSC 2000:*53Z05 Appl. of differential geometry to physics83C47 Quantum field theory on curved space-times

Keywords: nearly autoparallel map; conformal rescaling; nearly conformal map; quantum gravity


Zbl 0832.53071Văcaru, Sergiu I.Nearly autoparallel maps and conservation laws on curved spaces. (English)[J] Rom. J. Phys. 39, No. 1, 37-52 (1994). ISSN 1221-146X

The author presents a criterion for when it is possible to give an equivalent reformulation of theEinstein gravitational field equations in arbitrarily given spaces. The study is obtained by using themethod of nearly auto-parallel maps of generalized affine metric and bundle spaces. Geometricalconstructions are realized for Riemann-Cartan spaces and applications are considered only for theEinstein theory.\par Using the metricity conditions on Riemann-Cartan space, the components of the basic connection $\Gamma^a_{bc}$ are given in terms of the metric $g_{ab}$ and thecotorsion tensor $T^a_{bc}$, and this represents a generalization of the Einstein relation [see

{\it A. Einstein}, The meaning of relativity, especially ``Relativistic theory of the non-symmetricfield'', (5th ed.), (Princeton 1955; Zbl 0067.20404 )]. The torsion tensor is defined and thesymmetric affine is introduced. Thus, on Riemann-Cartan space three connection structures aredefined. Using conformal transformations of the metric one obtains the deformation tensor.\par OnEinstein-Cartan space the author considers two classes of curves which are reduced to geodesicson corresponding (pseudo)-Riemannian spaces when the torsion is equal to zero.\par Furthermore,a classification of Einstein-Cartan spaces is presented.[G.G.Vrănceanu (Bucureşti) ]

MSC 2000:*83C40 Groups of motions, etc.53C80 Appl. of global differential geometry to physics


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83D05 Relativistic gravitational theories other than Einstein's

Keywords: Einstein-Cartan space; Einstein equations; Riemann-Cartan spaceCitations: Zbl 0039.38802 ; Zbl 0050.21208 ; Zbl 0067.20404


Zbl 0829.53070Văcaru, Sergiu I. ; Ostaf, S. ; Goncharenko, Yu.( Gonchiarenko, Yu. )Nearly autoparallel maps and modelling of field interactions. (English)[J] Rom. J. Phys. 39, No.3-4, 199-211 (1994). ISSN 1221-146X

This paper is devoted to the analysis of Yang-Mills and Einstein theories on the basis of the$N$-maps transforms of bundle spaces. The theory of nearly geodesic mappings ($ng$-maps) foraffine connected spaces of {\it N. S. Sinyukov} [Geodesic mappings of Riemannian spaces,Moskva: Nauka (1979; Zbl 0637.53020 )], as well as of nearly autoparallel maps (na-maps) forgeneralized affine metric spaces of Vacaru et al. have been extended to the case of maps withsplitting of the bundle connection.\par This leads to a new way to find solutions of Yang-Mills andEinstein equations that sometimes may be reduced even to algebraic matrix equations.[ I.Gottlieb (Iaşi) ]

MSC 2000:*53Z05 Appl. of differential geometry to physics83C60 Spinor and twistor methods in general retativity

Keywords: Yang-Mills equations; splitting; bundle connection; Einstein equationsCitations: Zbl 0637.53020


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