Abstract:
Previously Chicone, Latushkin and Montgomery-Smith [\textbf{Comm. Math. Phys. \textbf{173},(1995)}] have investigated the spectrum of the dynamo operator for an ideally conducting fluid. More recently, Tang and Boozer [{\textbf{Phys. Plasmas (2000)}}], have investigated the anisotropies in magnetic field dynamo evolution, from finite-time, Lyapunov exponents, giving rise to a Riemann metric tensor, in the Alfven twist in magnetic flux tubes (MFTs). In this paper one investigate the role of Perelman Ricci flows constraints in twisted magnetic flux tubes, where the Lyapunov eigenvalue spectra for the Ricci tensor associated with the Ricci flow equation in MFTs leads to a finite-time Lyapunov exponential stretching along the toroidal direction of the tube and a contraction along the radial direction of the tube. It is shown that in the case of MFTs, the sectional Ricci curvature of the flow, is negative as happens in geodesic flows of Anosov type. Ricci flows constraints in MFTs substitute the Thiffeault and Boozer [\textbf{Chaos}(2001)] have vanishing of Riemann curvature constraint on the Lyapunov exponential stretching of chaotic flows. Gauss curvature of the twisted MFT is also computed and the contraints on a negative Gauss curvature are obtained.

Abstract:
Previously Casetti, Clementi and Pettini [\textbf{Phys.Rev.E \textbf{54},6,(1996)}] have investigated the Lyapunov spectrum of Hamiltonian flows for several Hamiltonian systems by making use of the Riemannian geometry. Basically the Lyapunov stability analysis was substituted by the Ricci sectional curvature analysis. In this report we apply Pettini's geometrical framework to determine the potential energy of a twisted magnetic flux tube, from its curved Riemannian geometry. Actually the Lyapunov exponents, are connected to a Riemann metric tensor, of the twisted magnetic flux tubes (MFTs). The Hamiltonian flow inside the tube is actually given by Perelman Ricci flows constraints in twisted magnetic flux tubes, where the Lyapunov eigenvalue spectra for the Ricci tensor associated with the Ricci flow equation in MFTs leads to a finite-time Lyapunov exponential stretching along the toroidal direction of the tube and a contraction along the radial direction of the tube. The Jacobi equation for the MFTs is shown to have a constant sectional Ricci curvature which allows us to compute the Jacobi-Levi-Civita (JLC) geodesic deviation for the spread of lines on the tube manifold and chaotic action through the greatest of its Lyapunov exponents. By analyzing the spectra of the twisted MFT, it is shown that the greater exponent is positive and proportional to the random radial flow of the tube, which allows the onset of chaos is guaranted. The randomness in the twisted flow reminds a discussed here is similar of a recent work by Shukurov, Stepanov, and Sokoloff on dynamo action on Moebius flow [\textbf{Phys Rev E 78 (2008)}]. The dynamo action in twisted flux tubes discussed here may also serve as model for dynamo experiments in laboratory.

Abstract:
Ricci and sectional curvatures of twisted flux tubes in Riemannian manifold are computed to investigate the stability of the tubes. The geodesic equations are used to show that in the case of thick tubes, the curvature of planar (Frenet torsion-free) tubes have the effect ct of damping the flow speed along the tube. Stability of geodesic flows in the Riemannian twisted thin tubes (almost filaments), against constant radial perturbations is investigated by using the method of negative sectional curvature for unstable flows. No special form of the flow like Beltrami flows is admitted, and the proof is general for the case of thin tubes. It is found that for positive perturbations and angular speed of the flow, instability is achieved, since the sectional Ricci curvature of the twisted tube metric is negative.

Abstract:
Here, an analytical version of numerical results is obtained in case of considering the laminar non-turbulent limit, of a twisted Riemannian thin flux tube. It is shown that the magnetic field is amplified, when electric current helicity and Riemann curvature are both negative. Thus spaces of positive and negative Riemannian curvatures seems to support dynamo action inside the torus, and not only negative Riemannian curvature surfaces as happens in 2D dynamos. New features appear in ${\alpha}^{2}$-dynamo twisted flow, using the approximation of thin tubes flux tubes. These solutions are obtained in the resonant profile of the toroidal and poloidal frequencies modes of the dynamo force-free flow.

Abstract:
Investigation of the eigenvalue spectra of dynamo solutions, has been proved fundamental for the knowledge of dynamo physics. Earlier, curvature-folding relation on dynamos in Riemannian spaces has been investigated [PPL 2008]. Here, analytical solutions representing general turbulent dynamo filaments are obtained in resistive plasmas. Turbulent diffusivity with vanishing kinetic helicity yields a fast mode for a steady dynamo eigenvalue. The magnetic field lays down on a local frame 2 plane along the filaments embedded in a 3D plasma. Curvature effects plays the role of folding in fast magnetic dynamos. In the present examples, plasma equipartition between normal and binormal components of the magnetic field components is considered. In the opposite case, oscillatory, purely imaginary, branches of the spectrum are found in dynamo manifold. Degenerate eigenvalues, are obtained when the dynamo growth rate coincides with the filaments curvature. Spectra of dynamo obtained are similar to the fast dynamo solution obtained by Arnold on a compact torus. Dynamo experiments making use of this kind of solution have been investigated by Shukurov et al [Phys Rev E, 2008] with Perm liquid sodium experiments on a circular torus implications. Another example of dynamo plasma is given by the Arnold-Beltrami-Childress [ABC] twisted magnetic flux tubes with stagnation points.

Abstract:
Any smooth surface in R^3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R^2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then the geodesic flow of the corresponding surface is Anosov. We apply this result to the theory of mechanical linkages and their dynamics: we provide a new example of a simple linkage whose physical behavior is Anosov. For the first time, the edge lengths of the mechanism are given explicitly.

Abstract:
Recently Guenther et al the globally diagonalized ${\alpha}^{2}$ dynamo operator spectrum [J Phys A 2007) in mean field media, and its Krein space related perturbation theory [J Phys A 2006). Earlier, an example of fast dynamos in stretch shear and fold Anosov maps have been given by Gilbert [PRSA [1993)). In this paper, analytical solutions representing general turbulent dynamo filaments are obtained in resistive plasmas. When turbulent diffusivity is present and kinetic helicity vanishes, a fast dynamo mode is obtained, and the Anosov eigenvalue obtained. The magnetic field lays down on a Frenet 2 plane along the filaments embedded in a 3D flow. Curvature effects on fast dynamo are also investigate. In case of weak curvature filaments the one dimensional manifolds in plasmas present a fast dynamo action. A parallel result has been obtained by Chicone et al [Comm Math Phys), in the case fast dynamo spectrum in two dimensional compact Riemannian manifolds of negative constant curvature, called Anosov spaces. While problems of embedding may appear in their case here no embedding problems appear since the one dimensional curved plasmas are embedded in three dimensional Euclidean spaces. In the examples considered here, equipartion between normal and binormal components of the magnetic field components is considered. In the opposite case, non Anosov oscillatory, purely imaginary, branches of the spectrum are found in dynamo manifold. Negative constant curvature non-compact $\textbf{H}^{2}$ manifold, has also been used in one-component electron 2D plasma by Fantoni and Tellez (Stat. Phys, (2008))

Abstract:
Riemann and sectional curvatures of magnetic twisted flux tubes in Riemannian manifold are computed to investigate the stability of the plasma astrophysical tubes. The geodesic equations are used to show that in the case of thick magnetic tubes, the curvature of planar (Frenet torsion-free) tubes have the effect ct of damping the flow speed along the tube. Stability of geodesic flows in the Riemannian twisted thin tubes (almost filaments), against constant radial perturbations is investigated by using the method of negative sectional curvature for unstable flows. No special form of the flow like Beltrami flows is admitted, and the proof is general for the case of thin magnetic flux tubes. In the magnetic equilibrium state, the twist of the tube is shown to display also a damping effect on the toroidal velocity of the plasma flow. It is found that for positive perturbations and angular speed of the flow, instability is achieved, since the sectional Ricci curvature of the magnetic twisted tube metric is negative. Solar flare production may appear from these geometrical instabilities of the twisted solar loops.

Abstract:
new examples of the theory recently proposed by ricca [pra(1991)] on the generalization of da rios-betchov intrinsic equations on curvature and torsion of classical non-riemannian vortex higher-dimensional string are given. in particular we consider applications to 3-dimesional fluid dynamics, including the case of a twisted flux tube and the fluid rotation. in this case use is made of da rios equation to constrain the fluid. integrals on the cartan connection are shown to be related to the integrals which represent the total frenet torsion and total curvature. by analogy with the blue phases twisted tubes in liquid crystals, non-riemannian geometrical formulation of the twisted flux tube in fluid dynamics is obtained. a theorem by ricca and moffatt on invariant integrals for the frenet curvature is used to place limits on the cartan integrals. the stationary incompressible flow case is also addressed in the non-riemannian case where cartan torsion scalars are shown to correspond to abnormalities of the congruence. geodesic motion is shown to be torsionless. vorticity is shown to be expressed in terms of abnormalities of the congruence, which is analogous to the result recenly obtained [garcia de andrade,prd(2004)], where the vorticity of the superfluid plays the role of cartan contortion vector in the context of analog gravity.

Abstract:
A plasma loop twisted Riemannian model is applied to torus dynamos twisted flows it leading to a slow dynamo such as in Moebius strip dynamo, recently considered by Shukurov, Stepanov and Sokoloff [Phys. Rev. \textbf{E 78},025301,(2008)] to modelling Perm dynamo torus in liquid sodium. Since diffusion and advection (stretching), are competing effects for dynamo action, plasma resistivity term is shown to be proportional to loops Riemann curvature (folding). Shukurov et al, also showed that based on Ponomarenko dynamo, a broader torus channel produces a better dynamo. These results agree with Schekochihin et al [Phys Rev \textbf{E} (2002)] where random filamentary magnetic fields are strengthen by curvature. Analysis of spectrum of chaotic fast dynamos, shows that Riemann curvature acts as a damping, since growth magnetic field rate is inversely proportional to Riemann curvature. Comparison with general relativistic MHD dynamo equation, shows that the Ricci tensor, which is a contraction of the Riemann tensor also appears in the diffusion term. Curvature of plasma loop is ${{R^{1}}_{212}}|_{\textbf{Plasma}}\approx{5.6{\times}10^{-19}m^{-2}}$, while for Perm torus is certainly higher. Thus slow dynamos are favoured in dynamo laboratories rather than in plasma loops. It is shown that the curvature-stretching flux rope dynamo coupling energy, coincides with the minimum twist energy ${\epsilon}_{\textbf{twist}}\approx{10^{30}TeV}$ stored in flux ropes. Torus flux tubes around black-holes remain in the order of $2MeV$ and GBR are around $10^{52}TeV$. Since the ${R^{1}}_{212}$ is negative, inflexionary flux tubes fast dynamos may be responsible for this CME mechanism in UHF plasma loops.