International Journal of Geometric Methods in Modern Physics

Based on the extended Harrison and Estabrook's differential form method, we obtain the Lie symmetries of two (2+1)-dimensional Toda-like lattices from two different sets of differential forms, respectively. Moreover it is shown that, for each lattice, the determining equations for the two sets give the same symmetries; and the set of differential forms for the lower-dimensional space can make the computation for finding symmetries simpler than the other.

This paper aims to introduce new classes of two-dimensional quasi-Einstein Riemannian manifolds, endowed with generalized Poincaré metrics.

The Mishchenko–Fomenko theorem on action-angle coordinates is extended to time-dependent superintegrable Hamiltonian systems.

We show that for any fixed point P₀ on a Riemann surface Σ the distinct realizations of cocycles in correspond to the natural appearances of the standard Heisenberg vertex operator algebra Π(P₀) and to the commutative Heisenberg vertex operator algebra Π⁰(P₀), respectively.

In a previous paper we have sketched a geometrical model for ferromagnetic crystals within a geometrized framework of simple materials with internal variables. Using a model of magnetizable media investigated by Maugin and coworkers, in the state vector the magnetization vector together with its space gradient (subjected to an evolution equation depending on both internal and external magnetic field) were included. In this paper we deepen the study of this geometrical model. In particular, from the exploitation of Clausius–Duhem inequality the entropy production and the state laws are calculated explicitly and starting from one appropriate dynamical system the expression for the entropy function and necessary conditions for its existence are got. Finally, using a Maugin's technique, the heat equation in the first and second form is derived.

Some dynamical and geometrical properties of controls dynamic for a drift-free left invariant control system from the Poisson geometry point of view are described. The integrability of such system are also studied.

In this study Fermi–Walker derivative and according to the derivative Fermi–Walker parallelism and non-rotating frame concepts are given for some frames. First, we get the Frenet frame, the Darboux frame, the Bishop frame for any curve in Euclid space. Fermi–Walker derivative and non-rotating frame being conditions are analyzed for each of the frames along the curve. Then we proved the Frenet frame is non-rotating frame along the plane curves. Darboux frame which is a non-rotating frame along the line of curvature. Then we proved the Bishop frame is a non-rotating frame along the all curves.

Partially supersymmetric intersecting (non-marginal) composite M-brane solutions defined on the product of Ricci-flat manifolds M₀ × M₁ × ⋯ × M_n in D = 11 supergravity are considered and formulae for fractional numbers of unbroken supersymmetries are derived for the following configurations of branes: M2 ∩ M2, M2 ∩ M5, M5 ∩ M5 and M2 ∩ M2 ∩ M2. Certain examples of partially supersymmetric configurations are presented.

Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the four-dimensional semiregular polytope snub 24-cell and its symmetry group (W(D₄)/C₂): S₃ of order 576 are obtained from the quaternionic representation of the Coxeter–Weyl group W(D₄). The symmetry group is an extension of the proper subgroup of the Coxeter–Weyl group W(D₄) by the permutation symmetry of the Coxeter–Dynkin diagram D₄. The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector Λ = (τ, 1, τ, τ) or on the vector Λ = (σ, 1, σ, σ) in the Dynkin basis with and . The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasiregular four-dimensional polytope invariant under the Coxeter–Weyl group W(F₄). Each vertex of the new polytope is shared by one cube and three truncated octahedra. Dual of the snub 24-cell is also constructed. Relevance of these structures to the Coxeter groups W(H₄) and W(E₈) has been pointed out.

We describe a program for developing a canonical gravity in 2 + 2 dimensions (two time and two space dimensions). Our procedure is similar to the usual canonical gravity but with two times rather than just one time. Our work may be of particular interest as an alternative approach to loop quantum gravity in 2 + 2 dimensions.

Deformations of the canonical spectral triples over the n-dimensional torus are considered. These deformations have a discrete dimension spectrum consisting of non-integer values less than n. The differential algebra corresponding to these spectral triples is studied. No junk forms appear for non-vanishing deformation parameter. The action of a scalar field in these spaces is considered, leading to non-trivial extra structure compared to the integer-dimensional cases, which does not involve a loss of covariance. One-loop contributions are computed leading to finite results for non-vanishing deformation.

In this note an interpretation of Riemann's zeta function is provided in terms of an ℝ-equivariant L²-index of a Dirac–Ramond type operator, akin to the one on (mean zero) loops in flat space constructed by the present author and T. Wurzbacher. We build on the formal similarity between Euler's partitio numerorum function (the S¹-equivariant L²-index of the loop space Dirac–Ramond operator) and Riemann's zeta function. Also, a Lefschetz–Atiyah–Bott interpretation of the result together with a generalization to M. Lapidus' fractal membranes are also discussed. A fermionic Bost–Connes type statistical mechanical model is presented as well, exhibiting a "phase transition at (inverse) temperature β = 1", which also holds for some "well-behaved" g-prime systems in the sense of Hilberdink–Lapidus.

We discuss in a critical way the physical foundations of geometric structure of relativistic theories of gravity by the so-called Ehlers–Pirani–Schild formalism. This approach provides a natural interpretation of the observables showing how relate them to General Relativity and to a large class of Extended Theories of Gravity. In particular we show that, in such a formalism, geodesic and causal structures of space-time can be safely disentangled allowing a correct analysis in view of observations and experiment. As specific case, we take into account the case of f(R)-gravity.

The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A novel geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpretation of the symmetric product yields two different applications. First, an intrinsic proof is provided of the fact that the distributions closed under the symmetric product are exactly those distributions invariant under the geodesic flow. Second, some applications in geometric control theory for mechanical systems are clarified.

We develop a Hamilton–Jacobi theory for singular Lagrangian systems in the Skinner–Rusk formalism. Comparisons with the Hamilton–Jacobi problem in the Lagrangian and Hamiltonian settings are discussed.