Differential equations of timelike geodesic surfaces in Riemannian space-time

Differential equations of timelike geodesic surfaces in Riemannian space-time are derived. These equations describe the motion of areal objects the internal parts of which are not affected by the total external force. Gravitational forces do not belong to external forces. They are taken into account in the geometrical objects of space-time.     

Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely:

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they admit geodesically equivalent metrics;     

Some topological invariants for three-dimensional flows

We deal here with vector fields on three manifolds. For a system with a homoclinic orbit to a saddle-focus point, we show that the imaginary part of the complex eigenvalues is a conjugacy invariant. We show also that the ratio of the real part of the complex eigenvalue over the real one is invariant under topological equivalence. For a system with two saddle-focus points and an orbit connecting the one-dimensional invariant manifold of those points, we compute a conjugacy invariant related to the eigenvalues of the vector field at the singularities. (c) 2001 American Institute of Physics.     

Conformal invariants for determinants of laplacians on Riemann surfaces

For a Riemann surface with smooth boundaries, conformal (Weyl) invariant quantities proportional to the determinant of the scalar Laplacian operator are constructed both for Dirichlet and Neumann boundary conditions. The determinants are defined by zeta function regularization. The other quantities in the invariants are determined from metric properties of the surface. As applications explicit representations for the determinants on the flat disk and the flat annulus are derived.     

Non-integrability of geodesic flow on certain algebraic surfaces

This Letter addresses an open problem recently posed by V. Kozlov: a rigorous proof of the non-integrability of the geodesic flow on the cubic surface xyz=1$xyz=1$. We prove this is the case using the Morales–Ramis theorem and Kovacic algorithm. We also consider some consequences and extensions of this result.     

Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus

In this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of Riemannian metrics on the two-dimensional torus. Also our method gives a new proof of Birkhoff's theorem for twist maps of the cylinder. We prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the Aubry-Mather set with this rotation number.     

Complexifications of geodesic flows and adapted complex structures

The existence of adapted complex structures for real-analytic Riemannian manifolds is examined under the point view of complexifications of geodesic flows. If the geodesic flow can be complexified to a complete holomorphic flow a sufficient criterion is given as to when a domain in the tangent bundle of the Riemannian manifold is a maximal domain of definition of an adapted complex structure. This criterion allows to determine maximal domains of definition for adapted complex structures of Berger spheres and of the Heisenberg group.     

On quadratic first integrals of the geodesic equations for type {22} spacetimes

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.     

Integration of some examples of geodesic flows via solvable structures

Solvable structures are particularly useful in the integration by quadratures of ordinary differential equations. Nevertheless, for a given equation, it is not always possible to compute a solvable structure. In practice, the simplest solvable structures are those adapted to an already known system of symmetries. In this paper we propose a method of integration which uses solvable structures suitably adapted to both symmetries and first integrals. In the variational case, due to Noether theorem, this method is particularly effective as illustrated by some examples of integration of the geodesic flows.     

Cuspidal edges for elastic wave surfaces for cubic crystals

The paper deals with a detailed numerical study of the sections of the inverse and ray velocity surfaces for cubic crystals. The figures for the sections of the inverse and ray surfaces by the (001) and (110) planes have been plotted for over 65 crystals and from these, the nature of the cuspidal edges has been discussed. Typical graphs of the inverse and ray surfaces have been given. The parameters characterising the dimensions of the cusps have been tabulated. It is shown that the A-15 compounds exhibit very unusual and interesting wave surfaces at temperatures below superconducting critical temperatures.     

Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows

This paper represents part of a program to understand the behavior of topological entropy for Anosov and geodesic flows. In this paper, we have two goals. First we obtain some regularity results forC ¹ perturbations. Second, and more importantly, we obtain explicit formulas for the derivative of topological entropy. These formulas allow us to characterize the critical points of topological entropy on the space of negatively curved metrics.Partially supported by NSF grant DMS-8514630Chaim Weizmann Research Fellow and NSF postdoctoral Research Fellow     

Perturbations of geodesic flows on surface of constant negative curvature and their mixing properties

We consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non-perturbed ones. One condition says that the Hamilton-Jacobi equation (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the SL(2, ) group. We also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function, equal to the curvature of the surface divided by 2. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time (up to logarithms).Part of this work was performed while the first and third authors were in residence at the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, MN 55455, USASupported by the Mathematics Dept. of Princeton University of by Stiftung Volkswagenwerk through IHES, and IMA     

Integral invariants for non-barotropic flows in a four dimensional space time manifold

Properties of non-barotropic flows are described using Lie derivatives of differential forms in a Euclidean four dimensional space-time manifold. Vanishing of the Lie derivative implies that the corresponding physical quantity remains invariant along the integral curves of the flow. Integral invariants of non-barotropic perfect and viscous flows are studied using the concepts of relative and absolute invariance of forms. The four dimensional expressions for the rate of change of the generalized circulation, generalized vorticity flux, generalized helicity and generalized parity in the case of ideal and viscous non-barotropic flows are thereby obtained.     

First integrals versus configurational invariants and a weak form of complete integrability

A confusion over the concept of first integrals, which has been created in a recent paper by Hall [13] is clarified. The clear distinction between first integrals and functions which are first integrals only on a specific, fixed hypersurface is discussed. Hall's terminology of configurational invariants is adopted for the latter case. The possible relevance of knowing configurational invariants for a Hamiltonian system is illustrated by results concerning a weak form of the theory on complete integrability.     

First order flows for N=2 extremal black holes and duality invariants

We derive explicitly the superpotential W for the non-BPS branch of $N=2$ extremal black holes in terms of duality invariants of special geometry. Although this is done for a one-modulus case (the $t^3$ model), the example gives $Z\ne 0$ black holes and captures the basic distinction from previous attempts on the quadratic series (vanishing C tensor) and from the other $Z=0$ cases. The superpotential W turns out to be a non-polynomial expression (containing radicals) of the basic duality invariant quantities. These are the same which enter in the quartic invariant $I_4$ for $N=2$ theories based on symmetric spaces. Using the flow equations generated by W, we also provide the analytic general solution for the warp factor and for the scalar field supporting the non-BPS black holes.