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Helicity, a topological degree that measures the winding and linking of vortex lines, is preserved by ideal (barotropic) fluid dynamics. In the context of the Hamiltonian description, the helicity is a Casimir invariant characterizing a foliation of the associated Poisson manifold. Casimir invariants are special invariants that depend on the Poisson bracket, not on the particular choice of the Hamiltonian. The total mass (or particle number) is another Casimir invariant, whose invariance guarantees the mass (particle) conservation (independent of any specific choice of the Hamiltonian). In a kinetic description (e.g., that of the Vlasov equation), the helicity is no longer an invariant (although the total mass remains a Casimir of the Vlasov's Poisson algebra). The implication is that some "kinetic effect" can violate the constancy of the helicity. To elucidate how the helicity constraint emerges or submerges, we examine the fluid reduction of the Vlasov system; the fluid (macroscopic) system is a "sub-algebra" of the kinetic (microscopic) Vlasov system. In the Vlasov system, the helicity can be conserved if a special helicity symmetry condition holds. To put it another way, breaking helicity symmetry induces a change in the helicity. We delineate the geometrical meaning of helicity symmetry and show that for a special class of flows (the so-called epi-two-dimensional flows), the helicity symmetry is written as ∂γ = 0 for a coordinate γ of the configuration space. [ABSTRACT FROM AUTHOR]

This paper addresses pure gauge questions in the study of asymptotically de Sitter spacetimes. We construct global solutions to the eikonal equation on de Sitter, whose level sets give rise to double null foliations, and give detailed estimates for the structure coefficients in this gauge. We show two results that are relevant for the foliations used by the author in the context of the stability problem of the expanding region of Schwarzschild de Sitter spacetimes: (i) Small perturbations of round spheres on the cosmological horizons lead to spheres that pinch off at infinity. (ii) Globally well-behaved double null foliations can be constructed from infinity using a choice of spheres related to the level sets of a new mass aspect function. While (i) shows that in the above stability problem a final gauge choice is necessary, the proof of (ii) already outlines a strategy for the case of spacetimes with decaying, instead of vanishing, conformal Weyl curvature. [ABSTRACT FROM AUTHOR]

The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid that are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose several realizations of pre-rackoid structures: One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we show an implementation of the (pre-)rackoid in a three-dimensional topological sigma model. [ABSTRACT FROM AUTHOR]

The aim of this series of papers is to generalize the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this paper (Paper I), we propose a generalization of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalization by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves – dubbed Dodgson waves – is introduced and geometrically characterized. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows us to generalize the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike foliation by codimension one (A)dS Carroll manifolds. [ABSTRACT FROM AUTHOR]

We propose that the distinction between interacting and noninteracting integrable systems is characterized by the Onsager matrix. It being zero is the defining property of a noninteracting integrable system. To support our view, various classical and quantum integrable chains are discussed. [ABSTRACT FROM AUTHOR]

We study a noncommutative analog of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a “Lorentzian spectral triple,” which can also be viewed as the “reverse Wick rotation” of a product spectral triple. This construction of “Lorentzian spectral triples” fits well into the Krein space approach to noncommutative Lorentzian geometry. [ABSTRACT FROM AUTHOR]

We fulfilled a group foliation of the system of n-dimensional (n ≥ 2) Lame equations of the classical static theory of elasticity with respect to the infinite subgroup contained in normal subgroup of main group of this system. It permitted us to move from the Lame equations to the equivalent unification of two first-order systems: automorphic and resolving. We obtained a general solution of the automorphic system. This solution is an n-dimensional analogue of the Kolosov-Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. It turned out that in the two-dimensional and three-dimensional cases, which have a physical meaning, this system is conformally invariant, while the Lame equations admit only a group of similarities of the Euclidean space. This is a big success, since in the method of group foliation, resolving equations usually inherit Lie symmetries subgroup of the full symmetry group that was not used for the foliation. In the threedimensional case for the solutions of the resolving system, we found the general form of the transformations similar to the Kelvin transformation. These transformations are the consequence of the conformal invariance of the resolving system. In the threedimensional case with a help of the complex dependent and independent variables, the resolving system is written as a simple complex system. This allowed us to find non-trivial exact solutions of the Lame equations, which direct for the Lame equations practically impossible to obtain. For this complex system, all the essentially distinct invariant solutions of the maximal rank we have found in explicit form, or we reduced the finding of those solutions to the solving of the classical one-dimensional equations of the mathematical physics: the heat equation, the telegraph equation, the Tricomi equation, the generalized Darboux equation, and other equations. For the resolving system, we obtained double wave of a special type and double waves of the shear deformations in an elastic medium. By the obtained formulas for generating new solutions, these solutions give an infinite set of exact solutions of this complex system. By the three-dimensional analog of the Kolosov-Muskhelishvili formula, the found solutions produce an infinite-dimensional vector space of exact solutions of the Lame equations. [ABSTRACT FROM AUTHOR]

We find all orthogonal metrics where the geodesic Hamilton-Jacobi equation separates and the Riemann curvature tensor satisfies a certain equation (called the diagonal curvature condition).All orthogonal metrics of constant curvature satisfy the diagonal curvature condition. The metrics we find either correspond to a Benenti system or are warped product metrics where the induced metric on the base manifold corresponds to a Benenti system. Furthermore, we show that most metrics we find are characterized by concircular tensors; these metrics, called Kalnins-Eisenhart-Miller metrics, have an intrinsic characterization which can be used to obtain them on a given space. In conjunction with other results, we show that the metrics we found constitute all separable metrics for Riemannian spaces of constant curvature and de Sitter space. [ABSTRACT FROM AUTHOR]

Given a Riemannian 3-manifold (M3, g0) endowed with a unit vector field U0 that is tangent to a conformal foliation, we require that the pair extend to a space-time (M,G) endowed with a space-like unit vector field Ut in such a way that Ut simultaneously generates null geodesics and is tangent to a conformal foliation on space-like slices t = constant. The property that the foliation be conformal is intimately related to CR-geometry. [ABSTRACT FROM AUTHOR]

We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes. [ABSTRACT FROM AUTHOR]

We show that any spherically symmetric spacetime locally admits a maximal space-like slicing and we give a procedure allowing its construction. The designed construction procedure is based on purely geometrical arguments and, in practice, leads to solve a decoupled system of first-order quasi-linear partial differential equations. We have explicitly built up maximal foliations in Minkowski and Friedmann spacetimes. Our approach admits further generalizations and efficient computational implementation. As by-product, we suggest some applications of our work in the task of calibrating numerical relativity complex codes, usually written in Cartesian coordinates. [ABSTRACT FROM AUTHOR]

In this paper we give a frame for application of the averaging method to Bose-Einstein condensates (BECs) and obtain an abstract result upon the dynamics of BECs. Using the averaging method, we determine the location where the modulated amplitude waves (periodic or quasi-periodic) exist and obtain that all these modulated amplitude waves (periodic or quasi-periodic) form a foliation by varying the integration constant continuously. Compared with the previous work, modulated amplitude waves studied in this paper have nontrivial phases and this makes the problem become more difficult, since it involves some singularities. [ABSTRACT FROM AUTHOR]

Let Fn=(M,F) be a Finsler manifold and G be the Sasaki-Matsumoto metric on TM°. Bejancu and Farran ['Finsler geometry and natural foliations on the tangent bundle,' Rep. Math. Phys. 58, 131 (2006)] proved that Fn=(M,F) is a Riemannian manifold if and only if the Sasaki-Matsumoto metric G on TM° is bundlelike for the vertical foliation. Let Fn1+n2=(M1×fM2,F) be the warped product Finsler manifold. In this paper the warped Sasaki-Matsumoto metric *G is introduced for the warped product Finsler manifold, and it is shown if the warped function f is not a constant, then *G on TM° is bundlelike for the warped vertical foliation V*(TM°) if and only if F1n1=(M1,F1) and F2n2=(M2,F2) are Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We prove a global foliation result, using areal time, for T2 symmetric space-times with a positive cosmological constant. We then find a class of solutions that exhibit asymptotically velocity-term dominated behavior near the singularity. [ABSTRACT FROM AUTHOR]

As a preliminary to discussing the quantization of the foliation in a history form of general relativity, we show how the discussion in an earlier work [J. Math. Phys. 43, 3053 (2002)] of a history version of free, scalar quantum field theory can be augmented in such a way as to include the quantization of the unit-length, timelike vector that determines a Lorentzian foliation of Minkowski space-time. We employ a Hilbert bundle construction that is motivated by (i) discussing the role of the external Lorentz group in the existing history quantum field theory [J. Math. Phys. 43, 3053 (2002)] and (ii) considering a specific representation of the extended history algebra obtained from the multi-symplectic representation of scalar field theory. [ABSTRACT FROM AUTHOR]

We discuss Hamiltonian structures of the Gelfand-Dorfman complex of projectable vector fields and differential forms on a foliated manifold. Such a structure defines a Poisson structure on the algebra of foliated functions, and embeds the given foliation into a larger, generalized foliation with presymplectic leaves. In a socalled tame case, the structure is induced by a Poisson structure of the manifold. Cohomology spaces and classes relevant to geometric quantization are also considered. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of global properties of prescribed mean curvature (PMC) foliations in cosmological space–times with local U(1)×U(1) symmetry and matter described by the Vlasov equation. It turns out that these space–times admit a global foliation by PMC surfaces as well, but the techniques to achieve this goal are more complex than in the cases considered in Paper I [Henkel (2002)]. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Details the completeness of the foliation of the Reissner-Nordstrom (RN) space-times. Consideration of the inner horizon of the spacetime as simultaneous with the end of the compactified RN universe; Minimization of the area of the hypersurface for a given volume of the world tube.

Develops a Poisson structure for generalized Lotka-Volterra (GLV) systems. Translation of algebraic GLV matrix properties into the Poisson context; Consequences of implementing the GLV formalism structure on Poisson counterpart; Indication of symplectic foliation and the Darboux canonical representation.

Examine the structure and the geometry of nonsymmetric dynamically homogeneous spaces (types II and III). Admittance of autoparallel distribution; Foliation and characterization of the nonsymmetric dynamically homogeneous spaces; Existence of an appropriate coordinate system.

Studies the residual gauge freedom within the null quasi-spherical gauge for space-times admitting an expanding shear-free null foliation. Geometric nature of the residual coordinate freedom; Explicit construction of nontrivial NQS metric representing space-times; Application of the results in testing numerical evolution codes.

In this paper, we define lower dimensional volumes associated with sub-Dirac operators for foliations. In some cases, we compute these lower dimensional volumes. We also prove the Kastler-Kalau-Walze type theorems for foliations with or without boundary. As a corollary, we give an explanation of the gravitational action for the Robertson-Walker space [a, b] ×f M3. [ABSTRACT FROM AUTHOR]