  Recently Tang and Boozer [{\textbf{Phys. Plasmas (2000)}}], have investigated
the anisotropies in magnetic field dynamo evolution, from local Lyapunov
exponents, giving rise to a metric tensor, in the Alfven twist in magnetic flux
tubes (MFTs). Thiffeault and Boozer [\textbf{Chaos}(2001)] have investigated
the how the vanishing of Riemann curvature constrained the Lyapunov exponential
stretching of chaotic flows. In this paper, Tang-Boozer-Thiffeault differential
geometric framework is used to investigate effects of twisted magnetic flux
tube filled with helical chaotic flows on the Riemann curvature tensor. When
Frenet torsion is positive, the Riemann curvature is unstable, while the
negative torsion induces an stability when time $t\to{\infty}$. This enhances
the dynamo action inside the MFTs. The Riemann metric, depends on the radial
random flows along the poloidal and toroidal directions. The Anosov flows has
been applied by Arnold, Zeldovich, Ruzmaikin and Sokoloff [\textbf{JETP
(1982)}] to build a uniformly stretched dynamo flow solution, based on Arnold's
Cat Map. It is easy to show that when the random radial flow vanishes, the
magnetic field vanishes, since the exponential Lyapunov stretches vanishes.
This is an example of the application of the Vishik's anti-fast dynamo theorem
in the magnetic flux tubes. Geodesic flows of both Arnold and twisted MFT
dynamos are investigated. It is shown that a constant random radial flow can be
obtained from the geodesic equation. Throughout the paper one assumes, the
reasonable plasma astrophysical hypothesis of the weak torsion. Pseudo-Anosov
dynamo flows and maps have also been addressed by Gilbert [\textbf{Proc Roy Soc
A London (1993)}
