Encyclopedic Reference Gennadi A. SARDANASHVILY theoretician and mathematical physicist, Principal Research Scientist at the Department of Theoretical Physics, Moscow State University, Russia

G. Sardanashvily was born March 13, 1950, Moscow, Soviet Union. In 1967, he graduated from the Mathematical Superior Secondary School No.2 (Moscow) with a silver award and entered the Physics Faculty of Moscow State University (MSU). In 1973, he graduated with Honours Diploma from MSU (diploma work: "Finitedimensional representations of the conformal group"). He was a Ph.D. student of the Department of Theoretical Physics of MSU under the guidance of Professor D.D. Ivanenko in 1973 – 76. Since 1976 he holds research positions at the Department of Theoretical Physics of MSU: assistant research scientist (1976 – 86), research scientist (1987 – 96), senior research scientist (1997 – 99), Principal Research Scientist (since 1999). In 1989 – 2004 he also was a visiting professor at the University of Camerino, Italy.

2 Gennadi Sardanashvily attained his Ph.D. degree in physics and mathematics from MSU in 1980, with Dmitri Ivanenko as his supervisor (Ph.D. thesis: "Fibre bundle formalism in some models of field theory"), and his D.Sc. degree in physics and mathematics from MSU in 1998 (Doctoral thesis: "Higgs model of a classical gravitational field"). His research area covers geometric methods in field theory, quantum theory and mechanics; gauge gravitation theory. His outstanding achievements include: (i) Gauge gravitation theory on natural bundles where gravity is a classical Higgs field providing a world manifold with a Lorentz structure. (ii) Comprehensive geometric formulation of classical field where classical fields are represented by sections of fibre bundles, including: •

the differential calculus on graded fibred manifolds and cohomology of a variational bicomplex,

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Lagrangian formalism on fibre bundles and graded manifolds in terms of infiniteorder jets,

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generalized Noether theorems for reducible degenerate Lagrangian systems in homology terms,

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prequantum BRST-extended Lagrangian theory of fields, higher-order antifields and ghosts.

(iii) Covariant Hamiltonian field theory on polysymplectic manifolds. (iv) Classical Higgs field theory on composite bundles. (v) Geometric formulation of non-relativistic time-dependent mechanics in terms of fibre bundles over R, and its quantization in a form of geometric quantization of symplectic foliations. (vi) Geometric formulation of relativistic mechanics in terms of jets of one-dimensional submanifolds. (vii) Extension of the Liouville-Arnold, Nekhoroshev and Mishchenko-Fomenko theorems on integrable Hamiltonian systems to a general case of not necessary compact invariant submanifolds. In 1979 - 2011, he lectures on algebraic and geometrical methods in field theory at the Department of Theoretical Physics of MSU and, In 1989 - 2004, on geometric methods in field theory at University of Camerino (Italy). He is an author of the course "Modern Methods in Field Theory" (in Russ.) in five volumes.

3 Gennadi Sardanashvily published 24 books and more than 350 scientific articles. He was the founder and Managing Editor (2003 – 2013) of International Journal of Geometric Methods in Modern Physics (World Scientific, Singapore).

Brief exposition of main results Geometric formulation of classical field theory In contrast to the classical and quantum mechanics and quantum field theory, classical field theory, the only one that allows for a comprehensive mathematical formulation. It is based on representation of classical fields by sections of smooth fibre bundles. Lagrangian theory on fibre bundles and graded manifolds Because classical fields are represented by sections of fibre bundles, Lagrangian field theory is developed as Lagrangian theory on fibre bundles. The standard mathematical technique for the formulation of such a theory are jet manifolds of sections of fibre bundles. As is seen Lagrangian formalism of arbitrary finite order, it is convenient to develop this formalism on the Frechet manifold J ∞ Y of infinite order jets of a fibre bundle Y → X because of operations increasing order. It is formulated in algebraic terms of the variational bicomplex, not by appealing to the variation principle. The jet manifold J ∞ Y is endowed with the algebra of exterior differential forms as a direct limit of algebras exterior differential forms on jet manifolds of finite order. This algebra is split into the so-called variational bicomplex, whose elements include Lagrangians L, and one of its coboundary operator is the variational Euler – Lagrange operator δL . The kernel of this operator is the Euler – Lagrange equation. Cohomology of the variational bicomplex has been defined that results both in a global solution of the inverse variational problem (what Lagrangians L are variationally trivial, i.e. δL =0) and the global first variational formula dL= δL + d H J , which the first Noether theorem follows from. Construction of Lagrangian field theory involves consideration of Lagrangian systems of both even, submitted by the sections bundles, and odd Grassmann variables. Therefore, Lagrangian formalism in terms of the variational bicomplex has been generalized to graded manifolds. Generalized second Noether theorem for reducible degenerate Lagrangian systems In a general case of a reduced degenerate Lagrangian, the Euler – Lagrange operator obeys nontrivial Noether identities, which are not independent and are subject to nontrivial first-order Noether identities, in turn, satisfying second-order Noether identities, etc. The hierarchy of these Noether identities under a certain cohomology condition is described by the exact cochain complex, called the Kozul – Tate complex. Generalized second Noether theorem associates a certain cochain sequence with this complex. Its ascent operator, called the gauge operator, consists of a gauge symmetry

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of a Lagrangian and gauge symmetries of first and higher orders, which are parameterized by odd and even ghost fields. This cochain sequence and the Kozul – Tate complex of Noether identities fully characterize the degeneration of a Lagrangian system, which is necessary for its quantization..

Generalized first Noether theorem for gauge symmetries In the most general case of a gauge symmetry of a Lagrangian field system, it is shown that the corresponding conserved symmetry current is reduced to a superpotential, i. e., takes the form J µ = W µ + ∂ν U µν , J µ = W µ + ∂ν U µν , where W vanishes on the Euler – Lagrange equations. Lagrangian BRST field theory A preliminary step to quantization of a reducible degenerate Lagrangian field system is its so-called BRST extension. Such an extension is proved to be possible if the gauge operator is prolonged to a nilpotent BRST operator, also acting on ghost fields. In this case, the above-mentioned cochain sequence becomes a complex, called the BRST complex, and an original Lagrangian admits the BRST extension, depending on original fields, antifields, indexing the zero and higher order Noether identities, and ghost fields, parameterizing zero and higher order gauge symmetries. Differential geometry of composite bundles In a number of models of field theory and mechanics, one uses composite bundles Y → Σ → X , when sections of a fibre bundle Σ → X describe, e.g., a background field, Higgs fields or function of parameters. This is due to the fact that, given a section h of a fibre bundle Σ → X , the pull-back bundle h * Y → X is a subbundle of Y → X . The correlation between connections on bundles Y → X , Y → Σ , Σ → X and h * Y → X were established. As a result, given a connection A on a bundle Y → Σ , one introduces the so-called vertical covariant differential D on sections of a fibre bundle Y → X , such that its restriction to h * Y → X coincides with the usual covariant differential for a connection induced on h * Y → X by a connection A. For applications, it is important that a Lagrangian of a physical model considered on a composition bundle Y → Σ → X is factorized through a vertical covariant differential D. Classical theory of Hiigs fields Although spontaneous symmetry breaking is a quantum effect, it was suggested that, in classical gauge theory on a principal bundle P → X , it is characterized by a reduction of a structure Lie group G of this bundle to some of its closed subgroups Lie H. By virtue to the well-known theorem, such a reduction takes place if and only if the factor-bundle P / H → X admits a global section h, which is interpreted as a classical Higgs field. Let us consider a composite bundle P → P / H → X and a fibre bundle Y → P / H associated with an H-principal bundle P → P / H . It is a composite bundle Y → P / H → X whose sections describe a system of matter fields with an exact

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symmetry group H and Hiigs fields. This is Lagrangian theory on a composite fibre bundle Y → P / H → X . In particular, a Lagrangian of matter fields depends on Higgs fields through a vertical covariant differential defined by a connection on a fibre bundle Y → P / H . An example of such a system of matter and Higgs fields are Dirac spinor fields in a gravitational field. Covariant (polysymplectic) Hamiltonian formalism of classical field theory Application of symplectic Hamiltonian formalism of conservative classical mechanics to field theory leads to an infinite-dimensional phase space, when canonical variables are values of fields in any given instant. It fails to be a partner of Lagrangian formalism of classical field theory. The Hamilton equations on such a phase space are not familiar differential equations, and they are in no way comparable to the Euler – Lagrange equations of fields. For a field theory with first order Lagrangians, covariant Hamiltonian formalism on polysymplectic manifolds, when canonical momenta are correspondent to derivatives of fields relative to all space-time coordinates, was developed. Lagrangian formalism and covariant Hamiltonian formalism for field models with hyperregular Lagrangians only are equivalent. A comprehensive relation between these formalisms was established in the class of almost regular Lagrangians, which includes all the basic field models. Gauge gravitation theory, where a gravitational field is treated as the Higgs one, responsible for spontaneous breaking of space-time symmetries Since gauge symmetries of Lagrangians of gravitation theory are general covariant transformations, gravitation theory on a world manifold X is developed as classical field theory in the category of so called natural bundles over X. Examples of such bundles are tangent TX and cotangent T*X bundles over X, their tensor products and the bundle LX of linear frames in TX. The latter is a principal bundle with the structure group GL(4,R). The equivalence principle in a geometrical formulation sets a reduction of this structure group to the Lorentz SO(1,3) subgroup that stipulates the existence of a global section g of the factor-bundle LX / SO(1,3) → X , which is a pseudo-Riemannian metric, i.e., a gravitational field on X. It enables one to treat a metric gravitation field as the Higgs one. The obtained gravitation theory is the metric-affine one whose dynamic variables are a pseudo-Riemannian metric and general linear connections on X. The Higgs field nature of a gravitational field g is characterized the fact that, in different pseudo-Riemannian metrics, the representation dx µ → γ a of the tangent covectors dx µ by Dirac’s matrices γ a and, consequently, the Dirac operators, acting on spinor fields, are not equivalent. A complete system of spinor fields with the exact Lorentz group of symmetries and gravitational fields is described sections of a composite bundle S → LX / SO(1,3) → X where S → LX / SO(1,3) is spinor bundle associated with LX → LX / SO(1,3) .

Geometric formulation of Lagrangian and Hamiltonian non-autonomous mechanics in terms of fibre bundles Hamiltonian formulation of autonomous classical mechanics on symplectic manifolds is not applied to non-autonomous mechanics, subject to time-dependent transformations.

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that permits depending on the time of conversion. It was suggested to describe nonrelativistic mechanics in the complete form, admitting time-dependent transformations, as particular classical field theory on fibre bundles Q → R over the time axis R. However, it differ from classical field theory in that connections on fibre bundles Q → R over R are always flat and, therefore, are not dynamic variables. They characterize reference systems in non-relativistic mechanics. The velocity and phase spaces of nonrelativistic mechanics are the first order jet manifold J 1Q of sections of Q → R and the vertical cotangent bundle V*Q of Q → R . There has been developed a geometric formulation of Hamiltonian and Lagrangian non-relativistic mechanics with respect to an arbitrary reference frame and, in more general setting, of mechanics described by second order dynamic equations. Geometric formulation of relativistic mechanics in terms of one-dimensional submanifolds In contrast to non-relativistic mechanics, relativistic mechanics admits transformations of time, depending on spatial coordinates. It is formulated in terms of one-dimensional submanifolds of a configuration manifold Q, when the space of non-relativistic velocities is the first-order jet manifold J 11Q of one-dimensional submanifolds of a manifold Q, which Lagrangian formalism of relativistic mechanics is based on. The generalization of the Liouville – Arnold, Nekhoroshev and Mishchenko – Fomenko theorems on the "action-angle" coordinates for completely and partially integrable and superintegrable Hamiltonian systems to the case of non-compact invariant submanifolds.

Other published results •

Spinor representations of the special conformal group

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Topology of stable points of the renormalization group

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Homotopy classification of curvature-free gauge fields

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Mathematical model of a discrete space-time

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Geometric formulation of the equivalence principle

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Classification of gravitation singularities as singularities of space-time foliations

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The Wheeler-deWitt superspace of spatial geometries with topological transitions

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Gauge theory of the “fifth force” as space-time dislocations

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Generating functionals in algebraic quantum field theory as true measures in the duals of nuclear spaces

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Generalized Komar energy-momentum superpotentials in metric-affine and gauge gravitation theories

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Non-holonomic constraints in non-autonomous mechanics

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Differential geometry of simple graded manifolds

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The geodesic form of second order dynamic equations in non-relativistic mechanics

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Classical and quantum mechanics with time-dependent parameters on composite bundles

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Geometry of symplectic foliations

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Geometric quantization of non-autonomous Hamiltonian mechanics

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Bi-Hamiltonian partially integrable systems and the KAM theorem for them

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Non-autonomous completely integrable and superintegrable Hamiltonian systems

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Geometric quantization of completely integrable and superintegrable Hamiltonian systems in the “action-angle” variables

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The covariant Lyapunov tensor and Lyapunov stability with respect to timedependent Riemannian metrics

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Relative and iterated BRST cohomology

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Non-equivalent representations of the algebra of canonical commutation relations modelled on an infinite-dimensional nuclear space

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Generalization of the Serre – Swan theorem to non-compact and graded manifolds

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Definition of higher-order differential operators in non-commutative geometry

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Conservation laws in higher-dimensional Chern – Simons models

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Classical and quantum Jacobi fields of completely integrable systems

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Classical and quantum non-adiabatic holonomy operators for completely integrable systems

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Classical and quantum mechanics with respect to different reference frames

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Lagrangian and Hamiltonian theory of submanifolds

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Geometric quantization of Hamiltonian relativistic mechanics

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Supergravity as a supermetric on supermanifolds

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Noether identities for differential operators

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Differential operators on generalized functions

Top publications Monographs • • • • • • • • • •

G. Sardanashvily, O. Zakharov. Gauge Gravitation Theory (World Scientific, Singapore, 1992). G. Sardanashvily. Generalized Hamiltonian Formalism for Field Theory (World Scientific, Singapore, 1995). G. Giachetta, L. Mangiarotti, G. Sardanashvily. New Lagrangian and Hamiltonian Methods in Field Theory (World Scientific, Singapore, 1997). L. Mangiarotti, G. Sardanashvily. Gauge Mechanics (World Scientific, Singapore, 1998). L. Mangiarotti, G. Sardanashvily. Connections in Classical and Quantum Field Theory (World Scientific, Singapore, 2000). G. Giachetta, L. Mangiarotti, G. Sardanashvily. Geometric and Algebraic Topological Methods in Quantum Mechanics (World Scientific, Singapore, 2005). G. Giachetta, L. Mangiarotti, G. Sardanashvily. Advanced Classical Field Theory (World Scientific, Singapore, 2009). G. Giachetta, L. Mangiarotti, G. Sardanashvily. Geometric Formulation of Classical and Quantum Mechanics (World Scientific, Singapore, 2010). G. Sardanashvily, Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory (Lambert Academic Publishing, Saarbrucken, 2012). G. Sardanashviily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (Lambert Academic Publishing, Saarbrucken, 2013)

Articles • • • • • • • • • •

G. Sardanashvily. Gravity as a Goldstone field in the Lorentz gauge theory, Phys. Lett. A 75 (1980) 257 – 258. D. Ivanenko and G. Sardanashvily. Foliation analysis of gravitational singularities, Phys. Lett. A 91 (1982) 341 – 344. D. Ivanenko and G. Sardanashvily. The gauge treatment of gravity, Phys. Rep. 94 (1983) 1 – 45. G. Sardanashvily and O. Zakharov. On functional integrals in quantum field theory, Rep. Math. Phys. 29 (1991) 101 – 108. G. Sardanashvily. On the geometry of spontaneous symmetry breaking, J. Math. Phys. 33 (1992) 1546 – 1549. G. Sardanashvily and O. Zakharov. On application of the Hamilton formalism in fibred manifolds to field theory, Diff. Geom. Appl. 3 (1993) 245 – 263. G. Sardanashvily. Constraint field systems in multimomentum canonical variables, J. Math. Phys. 35 (1994) 6584 – 6603. G. Giachetta and G. Sardanashvily. Stress-energy-momentum of affine-metric gravity. Generalized Komar superportential, Class. Quant. Grav. 13 (1996) L67 – L71. G. Sardanashvily. Stress-energy-momentum tensors in constraint field theories, J. Math. Phys. 38 (1997) 847 – 866. G. Sardanashvily. Stress-energy-momentum conservation law in gauge gravitation theory, Class. Quant. Grav. 14 (1997) 1371 – 1386.

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G. Sardanashvily. Hamiltonian time-dependent mechanics, J. Math. Phys. 39 (1998) 2714 – 2729. G. Sardanashvily. Covariant spin structure, J. Math. Phys. 39 (1998) 4874 – 4890. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Nonholonomic constraints in timedependent mechanics, J. Math. Phys. 40 (1999) 1376 – 1390. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Covariant Hamiltonian equations for field theory, J. Phys. A 32 (1999) 6629 – 6642. L. Mangiarotti and G. Sardanashvily. On the geodesic form of second order dynamic equations, J. Math. Phys. 41 (2000) 835 – 844. L. Mangiarotti and G. Sardanashvily. Constraints in Hamiltonian time-dependent mechanics, J. Math. Phys. 41 (2000) 2858 – 2876. G. Sardanashvily. Classical and quantum mechanics with time-dependent parameters, J. Math. Phys. 41 (2000) 5245 – 5255. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Iterated BRST cohomology, Lett. Math. Phys. 53 (2000) 143 – 156. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Cohomology of the infinite-order jet space and the inverse problem, J. Math. Phys. 42 (2001) 4272 – 4282. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Covariant geometric quantization of nonreletavistic time-dependent mechanics, J. Math. Phys. 43 (2002) 56 – 68. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Geometric quantization of mechanical systems with time-dependent parameters, J. Math. Phys. 43 (2002) 2882 – 2894. E. Fiorani, G. Giachetta and G. Sardanashvily. Geometric quantization of time-dependent completely integrable Hamiltonian systems, J. Math. Phys. 43 (2002) 5013 – 5025. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Action-angle coordinates for timedependent completely integrable Hamiltonian systems, J. Phys. A 35 (2002) L439 – L445. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Geometric quantization of completely integrable Hamiltonian systems in the action-angle variables, Phys. Lett. A 301 (2002) 53 – 57. E. Fiorani, G. Giachetta and G. Sardanashvily. The Liouville – Arnold – Nekhoroshev theorem for non-compact invariant manifolds, J. Phys. A 36 (2003) L101 – L107. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Jacobi fields of completely integrable Hamiltonian systems, Phys. Lett. A 309 (2003) 382 – 386. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Bi-Hamiltonian partially integrable systems, J. Math. Phys. 44 (2003) 1984 – 1997. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Nonadiabatic holonomy operators in classical and quantum completely integrable systems, J. Math. Phys. 45 (2004) 76 – 86. D. Bashkirov, G. Giachetta, L. Mangiarotti and G. Sardanashvily. Noether's second theorem for BRST symmetries, J. Math. Phys. 46 (2005) 053517. D. Bashkirov, G. Giachetta, L. Mangiarotti and G. Sardanashvily. Noether's second theorem in a general setting. Reducible gauge theories, J. Phys. A 38 (2005) 5329 – 5344. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology, Commun. Math. Phys. 259 (2005) 103 – 128. D. Bashkirov, G. Giachetta, L. Mangiarotti and G. Sardanashvily. The antifield Koszul – Tate complex of reducible Noether identities, J. Math. Phys. 46 (2005) 103513. G. Sardanashvily, Geometry of classical Higgs fields, Int. J. Geom. Methods Mod. Phys. 3 (2006) 139 – 148. E. Fiorani and G. Sardanashvily. Noncommutative integrability on noncompact invariant manifolds, J. Phys. A 39 (2006) 14035 – 14042. G. Giachetta, L. Mangiarotti and G. Sardanashvily. Quantization of noncommutative completely integrable Hamiltonian systems, Phys. Lett. A 362 (2007) 138 – 142. G. Sardanashvily, Graded infinite order jet manifolds, Int. J. Geom. Methods Mod. Phys. 4 (2007) 1335 – 1362. E. Fiorani and G. Sardanashvily. Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds, J. Math. Phys. 48 (2007) 032901. L. Mangiarotti and G. Sardanashvily. Quantum mechanics with respect to different reference frames, J. Math. Phys. 48 (2007) 082104. G. Sardanashvily, Supermetrics on supermanifolds, Int. J. Geom. Methods Mod. Phys. 5 (2008) 271 – 286.

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D. Bashkirov, G. Giachetta, L. Mangiarotti and G. Sardanashvily. The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237 – 252. G. Sardanashvily, Classical field theory. Advanced mathematical formulation, Int. J. Geom. Methods Mod. Phys. 5 (2008) 1163 – 1189. G. Giachetta, L. Mangiarotti and G. Sardanashvily. On the notion of gauge symmetries of generic Lagrangian field theory, J. Math. Phys. 50 (2009) 012903. G. Sardanashvily, Gauge conservation laws in a general setting. Superpotential, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1046 – 1057. G. Sardanashvily, Superintegrable Hamiltonian systems with noncompact invariant submanifolds, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1391 – 1420. G. Sardanashvily, Relativistic mechanics in a general setting, Int. J. Geom. Methods Mod. Phys. 7 (2010) 1307 – 1319. G. Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869 – 1895. G. Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics, J. Geom. Mech. 4 (2012) 99 – 110. G. Sardanashvily, Differential operators on Schwartz distributions, Int. J. Geom. Methods Mod. Phys. 9 (2012) 1250062. G. Sardanashvily, Time-dependent superintegrable Hamiltonian systems, Int. J. Geom. Methods Mod. Phys. 9 (2012) 1220016 G. Sardanashvily, Graded Lagrangian formalism, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1350016 G. Sardanashvily, Geometric formulation of non-autonomous mechanics, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1350061