International Journal of Geometric Methods in Modern Physics

New kind of spinors are introduced on four-dimensional Lorentzian Spin^c manifolds in [1]. We define Dirac operator on such spinors. In [2] Seiberg–Witten-like equations are written down on Minkowski space. In the present work we generalize these equations to four-dimensional Lorentzian Spin^c-manifolds.

A contact version of a Laudenbach's engulfing theorem is proved. Some properties of the notions of contact displacement energy and contact Hofer–Zehnder capacities are presented and, under the condition of existence of a modified action selector on a contact manifold, we can prove some inequalities involving these invariants. These inequalities are similar to the ones obtained by Frauenfelder, Ginzburg and Schlenk, in the symplectic case.

We study the geometry of a sub-Riemannian manifold (M, HM, VM, g), where HM and VM are the horizontal and vertical distribution respectively, and g is a Riemannian extension of the Riemannian metric on HM. First, without the assumption that HM and VM are orthogonal, we construct a sub- Riemannian connection ▽ on HM and prove some Bianchi identities for ▽. Then, we introduce the horizontal sectional curvature, prove a Schur theorem for sub-Riemannian geometry and find a class of sub-Riemannian manifolds of constant horizontal curvature. Finally, we define the horizontal Ricci tensor and scalar curvature, and some sub-Riemannian differential operators (gradient, divergence, Laplacian), extending some results from geometry to the sub-Riemannian setting.

The present paper deals with an intrinsic investigation of the notion of a parallel π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a parallel π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change (with being a parallel π-vector field), is established. 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-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 concluded.

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

This paper focuses on the study of a deformation of Finsler metrics satisfying Ricci flow equation. We will prove that every deformation of Randers (Kropina)-metrics satisfying Ricci flow equation is Einstein, we will also show that a deformation of Einstein metrics with initial Randers (Kropina)-metrics remains Randers (Kropina). In other words the deformation of Randers (or Kropina)-metrics is exactly the fixed point of (un-normal and normal) Ricci flow equation.

We introduce a study of Riemannian manifold M = ℝ² endowed with a metric of diagonal type of the form , where g is a positive function, of C^∞-class, depending on the variable x² only. We emphasize the role of metric in determining manifolds having negative, null or positive sectional curvature. Within this framework, we find a wide class of gradient Ricci solitons (see, Theorems 4 and 7) and specialize these results to discuss some 2D and 4D case studies. The present study can be thought as a natural continuation of those included in monograph [22] by Constantin Udrişte, and to those in the research article [12] by Richard S. Hamilton (the result in Proposition 8 is precisely the famous "Hamilton cigar" in polar coordinates).

A partly original description of gauge fields and electroweak geometry is proposed. A discussion of the breaking of conformal symmetry and the nature of the dilaton in the proposed setting indicates that such questions cannot be definitely answered in the context of electroweak geometry.

Over the last few years, Slavnov has proposed a formulation of quantum Yang–Mills theory in the Coulomb gauge which preserves simultaneously manifest Lorentz invariance and gauge invariance of the ghost field Lagrangian. This paper presents in detail some of the necessary calculations, i.e. those dealing with the functional integral for the S-matrix and its invariance under shifted gauge transformations. The extension of this formalism to quantum gravity in the Prentki gauge deserves consideration.

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently [20]. We also provide several examples to illustrate our construction.

Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n) × U(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n = 2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU(2) × SU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.