Symmetry analysis and conservation laws of the geodesic equations for the Reissner-Nordström de Sitter black hole with a global monopole

This paper is devoted to the comprehensive analysis of the problem of symmetries and conservation laws for the geodesic equations of the Reissner-Nordström de Sitter (RNdS) black hole with a global monopole. For this purpose, the system of geodesic equations is determined and the corresponding classical Lie point symmetry operators are obtained. An optimal system of one dimensional subalgebras is constructed and a brief discussion about the algebraic structure of the Lie algebra of symmetries is presented. Also, the Noether symmetries of the geodesic Lagrangian is calculated. Finally, by applying two methods including Noether’s theorem and direct method the conservation laws associated to the system of geodesic equations are obtained.     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Geodesic symmetries of homogeneous Kähler Manifolds

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kähler metric onG/T does not have volume-preserving geodesic symmetries. From the Kähler/de Rham decomposition of a compact homogeneous Kähler manifold [8], our result extends to the invariant Kähler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kähler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.     

Local conservation laws,symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

In this paper, we consider a Kudryashov‐Sinelshchikov equation that describes pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles. We show that this equation is rich in conservation laws. These conservation laws have been found by using the direct method of the multipliers. We apply the Lie group method to derive the symmetries of this equation. Then, by using the optimal system of 1‐dimensional subalgebras we reduce the equation to ordinary differential equations. Finally, some exact wave solutions are obtained by applying the simplest equation method.     

Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion

We concentrate on Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion describing the growth of cell populations. First, we perform a complete symmetry classification of the equation, and then we find some interesting similarity solutions by means of the symmetries and the variable coefficient heat equation. Local dynamical behaviors are analyzed via the solutions for the growing cell populations. Second, we show that the conservation law multipliers of the equation take the form Λ=Λ(t,x,u), which satisfy a linear partial differential equation, and then give the general formula of conservation laws. Finally, symmetry properties of the conservation law are investigated and used to construct conservation laws of the reduced equations.     

Riemannian geometry of finite rank positive operators

A riemannian metric is introduced in the infinite dimensional manifold Σn of positive operators with rank n<∞ on a Hilbert space H. The geometry of this manifold is studied and related to the geometry of the submanifolds Σp of positive operators with range equal to the range of a projection p (rank of p=n), and Pp of selfadjoint projections in the connected component of p. It is shown that these spaces are complete in the geodesic distance.     

Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.     

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier–Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (\(1+1\))-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

     

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Group properties and conservation laws for nonlocal shallow water wave equation

Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.     

Conservation laws for one-layer shallow water wave systems

The problem of correspondence between symmetries and conservation laws for one-layer shallow water wave systems in the plane flow, axisymmetric flow and dispersive waves is investigated from the composite variational principle of view in the development of the study [N.H. Ibragimov, A new conservation theorem, Journal of Mathematical Analysis and Applications, 333(1) (2007) 311–328]. This method is devoted to construction of conservation laws of non-Lagrangian systems. Composite principle means that in addition to original variables of a given system, one should introduce a set of adjoint variables in order to obtain a system of Euler–Lagrange equations for some variational functional. After studying Lie point and Lie–Bäcklund symmetries, we obtain new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to shallow water wave systems. In particular, we obtain infinite local conservation laws and potential symmetries for the plane flow case.     

Topological rigidity theorems for open Riemannian manifolds

In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?ⁿ if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?ⁿ if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.     

A short proof of the generalized Bonnet theorem

In three‐dimensional Euclidean space E³, the Bonnet theorem says that a curve on a ruled surface in three‐dimensional Euclidean space, having two of the following properties, has also a third one, namely, it can be a geodesic, that it can be the striction line, and that it cuts the generators under constant angle. In this work, in n dimensional Euclidean space Eⁿ, a short proof of the theorem generalized for (k + 1) dimensional ruled surfaces by Hagen in 4 is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Resolvent and lattice points on symmetric spaces of strictly negative curvature

We study the asymptotics of the lattice point counting function for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group of motions in X, such that has finite volume. We show that as , for each . The constant corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions of the Laplacian, such that the eigenvalues are less than . Received: 4 January 1999     

On the connections between symmetries and conservation rules of dynamical systems

The strict connection between Lie point‐symmetries of a dynamical system and its constants of motion is discussed and emphasized through old and new results. It is shown in particular how the knowledge of the symmetry of a dynamical system can allow us to obtain conserved quantities that are invariant under the symmetry. In the case of Hamiltonian dynamical systems, it is shown that if the system admits a symmetry of a ‘weaker’ type (specifically, a λ or a Λ‐symmetry), then the generating function of the symmetry is not a conserved quantity, but the deviation from the exact conservation is ‘controlled’ in a well‐defined way. Several examples illustrate the various aspects. Copyright © 2012 John Wiley & Sons, Ltd.     

Translationally isotropic flat homogeneous manifolds with metric signature (n, 2)

In the following paper we classify the connected flat homogeneous spacesM of metric signature (n, 2) with translationally isotropic associated domain.U IR _s ⁿ whereIR _s ⁿ denotesIR ⁿ with the usual flat metric istranslationally isotropic if the set of all translations which leaveU invariant contains its perpendicular space. IfU is the image of the universal cover ofM under the development map thenU is called theassociated domain ofM.     

Weil-Petersson geometry of Teichmüller–Coxeter complex and its finite rank property

On a Teichmüller space, the Weil-Petersson metric is known to be incomplete. Taking metric and geodesic completions result in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson metric. We construct a geodesic completion of the Teichmüller space through the formalism of Coxeter complex with the Teichmüller space as its non-linear non-homogeneous fundamental domain. We then show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups.     

Diff(Rn) as a Milnor–Lie group

We describe a construction of the Lie group structure on the diffeomorphism group Diff( R ⁿ), modelled on the space D( R ⁿ, R ⁿ) of R ⁿ‐valued test functions on R ⁿ, in John Milnor's setting of infinite‐dimensional Lie groups. New tools are introduced to simplify this task. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)