International Journal of Geometric Methods in Modern Physics

In this paper we extend to the Fréchet setting the following well-known fact about finite-dimensional symplectic geometry: if a Lie group G acts on a symplectic manifold in a Hamiltonian fashion with momentum map μ, given x ∈ M the isotropy group G_x acts linearly on the tangent space in a Hamiltonian fashion, with momentum map the Taylor expansion of μ up to degree 2. We use this result to give a conceptual explanation for a formula of Donaldson in [Scalar curvature and projective embeddings. I, J. Differential Geom.59(3) (2001) 479–522], which describes the momentum map of the Hamiltonian infinitesimal action of the Lie algebra of the group of Hamiltonian diffeomorphisms of a closed integral symplectic manifold, on sections of its prequantum line bundle.

In this paper, for a Finsler manifold (M, F) with a Finsler metric g_ij(x, y) we shall consider a generalized Lagrange metrics (FGL-metrics) as the form *g_ij(x, y) = g_ij(x, y) + σ(x, y)B_i(x, y)B_j(x, y) on TM. Then we shall consider a Riemannian manifold (TM, *G) in which *G is a generalized Sasakian metric of *g on . Then we restrict the above FGL-metrics to a submanifold of , and show that it admits a GL-metric structure. Then we shall find a necessary and sufficient condition for this submanifold to be totally geodesic.

On a Riemannian almost product manifold (M, P, g), we consider a linear connection preserving the almost product structure P and the Riemannian metric g and having a totally skew-symmetric torsion. We determine the class of the manifolds (M, P, g) admitting such a connection and prove that this connection is unique in terms of the covariant derivative of P with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold (G, P, g) constructed by a Lie group G.

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)_n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)_n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)_n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)₄ spacetime manifolds.

In this paper we investigate the quantum cohomologies of symmetric products of Kähler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov–Witten invariant and relative Gromov–Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line ℙ¹, which is the k-dimensional complex projective space ℙ^k.

We review the Cauchy problem for f(R) theories of gravity, in metric and metric-affine formulations, pointing out analogies and differences with respect to General Relativity. The role of conformal transformations, effective scalar fields and sources in the field equations is discussed in view of the well-formulation and the well-position of the problem. Finally, criteria of viability of the f(R)-models are considered according to the various matter fields acting as sources.

It is shown that in the case of spontaneous breaking of the original gauge symmetry, a dynamic rearrangement of vacuum may lead to the formation of some anisotropic condensates. The appearance of such condensates causes the respective phase transitions in the geometric structure of space-time and creates a flat anisotropic, i.e. Finslerian event space. Actually there arises either a flat relativistically invariant Finslerian space with partially broken 3D isotropy, i.e. axially symmetric space, or a flat relativistically invariant Finslerian space with entirely broken 3D isotropy. The fact that any entirely anisotropic relativistically invariant Finslerian event space belongs to a 3-parameter family of such spaces gives rise to a fine structure of the respective geometric phase transitions. In the present paper the fine structure of the geometric phase transitions is studied by classifying all the metric states of the entirely anisotropic event space and the respective mass shell equations.

Para-Hermite–Einstein spacetimes are considered. It is shown that any para-Hermite–Einstein metric is locally defined by one nonlinear partial differential equation of the second order for one function.

In this paper, by using the nonlinear coherent states approach, we find a relation between the geometric structure of the physical space and the geometry of the corresponding projective Hilbert space. To illustrate the approach, we explore the quantum transition probability and the geometric phase in the curved space.

A new kind of spinors and Dirac operator are introduced on Kähler–Norden manifolds in [Spinors on Kähler–Norden manifolds, J. Nonlinear Math. Phys.17(1) (2010) 27–34]. In this work the square D² of the Dirac operator D is computed and a kind of Schrödinger–Lichnerowicz formula is obtained.

In this paper we address two problems concerning a family of domains M_Ω(μ) ⊂ ℂⁿ, called Cartan–Hartogs domains, endowed with a natural Kähler metric g(μ). The first one is determining when the metric g(μ) is extremal (in the sense of Calabi), while the second one studies when the coefficient a₂ in the Engliš expansion of Rawnsley ε-function associated to g(μ) is constant.

We present other examples illustrating the operator-theoretic approach to invariant integrals on quantum homogeneous spaces developed by Kürsten and the second author. The quantum spaces are chosen such that their coordinate algebras do not admit bounded Hilbert space representations and their self-adjoint generators have continuous spectrum. Operator algebras of trace class operators are associated to the coordinate algebras which allow interpretations as rapidly decreasing functions and as finite functions. The invariant integral is defined as a trace functional which generalizes the well-known quantum trace. We argue that previous algebraic methods would fail for these examples.

In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space of the model over the space X of basic variables. Extended constitutive surfaces Σ_S,κ are described as the Legendre submanifolds Σ_S of the space shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξ_K are tangent to the surfaces Σ_S,κ, introduce conformally Hamiltonian systems μξ_K where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the Coleman–Owen model of thermoelastic point.

The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group G = SO(3), acting canonically on three- and four-dimensional Euclidean spaces. We show that the property of the connection to be SO(3)-invariant allows us to find complete explicit description of all solutions of the SO(3)-metrizability equations.