International Journal of Geometric Methods in Modern Physics

On a strictly q-convex domain D in a Kähler manifold X, we obtain the solvability of the -problem for smooth forms and currents on boundaries of D. Moreover, we study the solvability of the -problem for extensible currents.

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

In this paper we will give a formula for computing conservation laws for a Hamiltonian system that admits non-Noether infinitesimal symmetry. The formula involves the differential operator associated with the dual Lefschetz operator corresponding to a symplectic form.

We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.

This paper aims to be a natural continuation of our very recent research, published in [Int. J. Geom. Meth. Mod. Phys.8(4) (2011) 783–796]. Using as geometric background a generalized Poincaré manifold, this work presents models of new explicit classes of manifolds within the class of gradient Ricci solitons. In Sec. 1 we shortly introduce a history of this topic to motivate its relevance both for theoretical and applied sciences, and in Sec. 2 our work contains its notations and assumptions. In Sec. 3 we give our framework and introduce the basic result, and in Sec. 4 we develop our theory. Finally, in Sec. 5, we conclude this paper and suggest possible further development. While the background in the first two sections is introductory, the theory in Secs. 3 and 4 is new as a whole, containing original results.

ν-classes are supergeometric generalization of the Chern classes. Associated to each superline bundle over supermanifold , ν-class is an element of the second cohomology group of M with coefficients in ℤ[ν] with ν² = 1.

The present paper deals with an intrinsic generalization of the conformal change and energy β-change on a Finsler manifold (M.L.), namely the energy β-conformal change ( with ; being a concurrent π-vector field and σ(x) is a function on M). The relation between the two Barthel connections Γ and , corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-conformal change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is obtained.

It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

We introduce geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry, and find geometrical characteristics associated to equations of Mathematical Physics. Also, we introduce a geometric study of some boundary problems. Throughout this work, as main tool we employed an adequate Riemannian Hessian structure, suggested in [Int. J. Geom. Meth. Mod. Phys.7(7) (2010) 1104–1113].

The paper is devoted to the achievements of two complementary multitime missions — to state a multitime Boussinesq PDE, as geometrical prolongation of the classical one, and to obtain its multitime soliton solutions. The advantages of geometrical ingredients capable to create this significant multitime PDE are clearly shown together with problems that were or should be overcome in the similar problems. The main goal of both missions is the investigation of multitemporal Boussinesq waves. Stressing the importance of the geometry on the multitime evolution phenomena, we add original results on multitime Boussinesq solitons and on Riccati ODEs of superior order.

In a paper due to [I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster parallel shape operator, Kodai Math. J.34 (2011) 352–366] we have shown that there does not exist a hypersurface in G₂(ℂ^m+2) with parallel shape operator in the generalized Tanaka–Webster connection (see [N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math.20 (1976) 131–190; S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc.314(1) (1989) 349–379]). In this paper, we introduce a new notion of generalized Tanaka–Webster 𝔇^⊥-parallel for a hypersurface M in G₂(ℂ^m+2), and give a characterization for a tube around a totally geodesic ℍ Pⁿ in G₂(ℂ^m+2) where m = 2n.

We compute the first cohomology of the ortosymplectic Lie superalgebra 𝔬𝔰𝔭(1|2) on the (1, 1)-dimensional real superspace with coefficients in the superspace 𝔇_{λ, μ} of bilinear differential operators acting on weighted densities (λ ∈ ℝ² and μ ∈ ℝ). This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on ℝℙ¹ acting on bilinear differential operators, Int. J. Geom. Meth. Mod. Phys.2(1) (2005) 23–40].

The collection of all projective vector fields on a Finsler space (M, F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra. A specific Lie sub-algebra of projective algebra of Randers spaces (called the special projective algebra) of non-zero constant S-curvature is studied and it is proved that its dimension is at most . Moreover, a local characterization of Randers spaces whose special projective algebra has maximum dimension is established. The results uncover somehow the complexity of projective Finsler geometry versus Riemannian geometry.

Four-dimensional A₄ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter–Weyl group W(A₄) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary W(A₄) orbit into three dimensions is made using the subgroup W(A₃). A generalization of the Catalan solids for 3D-polyhedra has been developed and dual polytopes of the uniform A₄ polytopes have been constructed.

In the preceding paper (arXiv: 1103.0329 [quant-ph]) we treated the Jaynes–Cummings model with dissipation and gave an approximate solution to the master equation for the density operator under the general setting by making use of the Zassenhaus expansion.

However, to obtain a compact form of the approximate solution (which is in general complicated infinite series) is very hard when an initial condition is given. To overcome this difficulty we develop another approach and obtain a compact approximate solution when some initial condition is given.

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler–Lagrange equations. Interesting applications as, for instance, a discrete derivation of the Euler–Poincaré equations for second-order Lagrangians and its application to optimal control of a rigid body, and of a Cosserat rod are shown at the end of the paper.

Every non-Riemannian (α, β)-metric on a manifold with dimension n ≥ 3 is semi-C-reducible. Thus the study on semi-C-reducible metrics will enhance our understanding on the physical meaning of (α, β)-metrics. In this paper, we study weakly Landsberg semi-C-reducible metrics with some non-Riemannian curvature properties. Then we find some conditions on semi-C-reducible manifolds under which the notions of isotropic Landsberg curvature and of isotropic mean Landsberg curvature are equivalent.