International Journal of Geometric Methods in Modern Physics

In this paper, we introduce generalized-quaternionic Kähler analogue of Lagrangian and Hamiltonian mechanical systems. Finally, the geometrical-physical results related to generalized-quaternionic Kähler mechanical systems are also given.

Let E → M be a holomorphic vector bundle over a compact Kähler manifold (M, ω) and let E = E₁ ⊕ ⋯ ⊕ E_m → M be its decomposition into irreducible factors. Suppose that each E_j admits a ω-balanced metric in Donaldson–Wang terminology. In this paper we prove that E admits a unique ω-balanced metric if and only if for all j, k = 1,…, m, where r_j denotes the rank of E_j and N_j = dim H⁰(M, E_j). We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety (M, ω) and we show the existence and rigidity of balanced Kähler embedding from (M, ω) into Grassmannians.

This paper has two aims. First, we show that the usual notion of umbilical maps between Riemannian manifolds does not work for Riemannian maps. Then we introduce a new notion of umbilical Riemannian maps between Riemannian manifolds and give a method on how to construct examples of umbilical Riemannian maps. In the second part, as a generalization of CR-submanifolds, holomorphic submersions, anti-invariant submersions, invariant Riemannian maps and anti-invariant Riemannian maps, we introduce semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds, give examples and investigate the geometry of distributions which are arisen from definition. We also obtain a decomposition theorem and give necessary and sufficient conditions for a semi-invariant Riemannian map to be totally geodesic. Then we study the geometry of umbilical semi-invariant Riemannian maps and obtain a classification theorem for such Riemannian maps.

In this paper we give a method to stabilize asymptotically the Lyapunov stable equilibrium states of a system describing the planar motions of an autonomous underwater vehicle.

The noncommutativity of a four-dimensional phase space is introduced from a purely symplectic point of view. We show that there is always a coordinate map to locally eliminate the gauge fluctuations inducing the deformation of the symplectic structure. This uses the Moser's lemma; a refined version of the celebrated Darboux theorem. We discuss the relation between the coordinates change arising from Moser's lemma and the Seiberg–Witten map. As illustration, we consider the quantum Hall systems on CP². We derive the action describing the electromagnetic interaction of Hall droplets. In particular, we show that the velocities of the edge field, along the droplet boundary, are noncommutativity parameters-dependents.

Let be a para-quaternionic Hermitian structure on the real vector space V. By referring to the tensorial presentation , we give an explicit description, from an affine and metric point of view, of main classes of subspaces of V which are invariantly defined with respect to the structure group of and respectively.

We engineer U(1)ⁿ Chern–Simons type theories describing fractional quantum Hall solitons (QHS) in 1 + 2 dimensions from M-theory compactified on eight-dimensional hyper-Kähler manifolds as target space of N = 4 sigma model. Based on M-theory/type IIA duality, the systems can be modeled by considering D6-branes wrapping intersecting Hirzebruch surfaces F₀'s arranged as ADE Dynkin Diagrams and interacting with higher-dimensional R-R gauge fields. In the case of finite Dynkin quivers, we recover well known values of the filling factor observed experimentally including Laughlin, Haldane and Jain series.

The massless supermultiplet of 11-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F₄ into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical Kaluza–Klein origin of this observation is proposed with internal space the Cayley plane, 𝕆P², and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions.

Let M be a C^∞ manifold, and let be the space of all smooth maps from [0, 1] to M. We investigate geometric structures on constructed from the geometric structures on M. In particular, we show that a generalized (almost) complex structure on M produce a generalized (almost) complex structure on .

By resorting to basic features of topological knot theory we propose a (classical) cryptographic protocol based on the "difficulty" of decomposing complex knots generated as connected sums of prime knots and their mutants. The scheme combines an asymmetric public key protocol with symmetric private ones and is intrinsically secure against quantum eavesdropper attacks.

Unitary matrices are the quantum gates in quantum computing. We study the question under which conditions is the commutator of two unitary matrices again a unitary matrix. We consider the general case and a complete solution for the 2 × 2 unitary matrices. We also consider Kronecker products and direct sums of unitary matrices.

In this paper, we introduce a Riemannian metric and a family of framed f-structures on the slit tangent bundle of a Finsler manifold Fⁿ = (M, F). Then we prove that there exists an almost contact structure on the tangent bundle, when this structure is restricted to the Finslerian indicatrix. We show that this structure is Sasakian if and only if Fⁿ is of positive constant curvature 1. Finally, we prove that (i) Fⁿ is a locally flat Riemannian manifold if and only if , (ii) the Jacobi operator is zero or commuting if and only if (M, F) have the zero flag curvature.

In this paper, we investigate the generalized Hyres–Ulam–Rassias stability and Bourgin-type superstability of higher derivations in non-Archimedean Banach algebras by using a version of fixed-point theorem.

A new relation among a class of optimal control systems and Lagrangian systems with symmetry is discussed. It will be shown that a family of solutions of optimal control systems whose control equation are obtained by means of a group action are in correspondence with the solutions of a mechanical Lagrangian system with symmetry. This result also explains the equivalence of the class of Lagrangian systems with symmetry and optimal control problems discussed in [1, 2].

The explicit realization of this correspondence is obtained by a judicious use of Clebsch variables and Lin constraints, a technique originally developed to provide simple realizations of Lagrangian systems with symmetry. It is noteworthy to point out that this correspondence exchanges the role of state and control variables for control systems with the configuration and Clebsch variables for the corresponding Lagrangian system.

These results are illustrated with various simple applications.

We analyze Lemaître–Tolman–Bondi models in presence of the cosmological constant Λ through the classical Weierstrass criterion. Precisely, we show that the Weierstrass approach allows us to classify the dynamics of these inhomogeneous spherically symmetric Universes taking into account their relationship with the sign of Λ.