International Journal of Geometric Methods in Modern Physics

As a generalization of isometric immersions and Riemannian submersions, Riemannian maps were introduced by Fischer [Riemannian maps between Riemannian manifolds, Contemp. Math.132 (1992) 331–366]. It is known that a real valued Riemannian map satisfies the eikonal equation which provides a bridge between physical optics and geometrical optics. In this paper, we introduce invariant and anti-invariant Riemannian maps between Riemannian manifolds and almost Hermitian manifolds as a generalization of invariant immersions and totally real immersions, respectively. Then we give examples, present a characterization and obtain a geometric characterization of harmonic invariant Riemannian maps in terms of the distributions which are involved in the definition of such maps. We also give a decomposition theorem by using the existence of invariant Riemannian maps to Kähler manifolds. Moreover, we study anti-invariant Riemannian maps, give examples and obtain a classification theorem for umbilical anti-invariant Riemannian maps.

In the present work we consider E is a Banach space, E* is its dual space and L(E) is the space of continuous linear operators from E to itself. A function x: ℝ → E is said to be a pseudo-solution of the equation

We consider composition and division algebras over the real numbers: We note two rôles for the group G₂: as automorphism group of the octonions and as the isotropy group of a generic three-form in seven dimensions. We show why they are equivalent, by means of a regular metric. We express in some diagrams the relation between some pertinent groups, most of them related to the octonions. Some applications to physics are also discussed.

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal.

We study a hierarchy of nonlinear autonomous systems of first ordinary differential equations. We show that it is completely integrable using the Painlevé test and finding the first integrals. We also show that it can be derived from the first integrals using Nambu mechanics. A corresponding dynamical system with chaotic behavior is also derived. A modified system with harmonic oscillators as first integrals is also considered. A connection with the Yang–Mills equation and the self-dual Yang–Mills equation is also discussed.

It is demonstrated that the Cartesian momenta and positions offer a proper description for the motion of particles constrained on the surface of a sphere. In classical limit, their mean values on the classical wave packet go over to classical quantities.

Consider a C¹ vector field X on a finite-dimensional manifold M and x_e an equilibrium point for the dynamics . We will prove that if I is a C² constant of motion for X such that x_e is also a critical point of I, then is a constant of motion for the linearized system . As a consequence we will give an instability result under the condition that is an indefinite quadratic form. If x_e is not a critical point of I, then we obtain as a constant of motion for the linearized system.

The geometric formulation of Hamilton–Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton–Jacobi problem with the symplectic structure defined from the Lagrangian function and the constraints is studied. The concept of complete solutions and their relationship with constants of motion, are also studied in detail. Local expressions using quasivelocities are provided. As an example, the nonholonomic free particle is considered.

The so-called Gell-Mann formula expresses the Lie algebra elements in terms of the corresponding Inönü–Wigner contracted ones. In the case of sl(n, ℝ) and su(n) algebras contracted w.r.t. so(n) subalgebras, the Gell-Mann formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell-Mann formula for sl(5, ℝ) and su(5) algebras, that is valid for all representations, is obtained in a group manifold framework of the SO(5) and/or Spin(5) group. Matrix elements of the sl(5, ℝ) noncompact operators, relevant for Affine Gauge Gravity and four-branes, are obtained for an arbitrary irreducible representation.

The Simpson's formula is obtained by approximating the integral of a function on some interval by the integral of the quadratic polynomial determined by the function. However, a multidimensional analogue of the formula has not been given as far as we know.

In this paper such a formula is given. Our formula is simple and beautiful, so it may be convenient in Mathematics or Mathematical Physics.

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature.