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Recently Vishik anti-fast dynamo theorem, has been tested againstnon-stretching flux tubes [Phys Plasmas 15 (2008)]. In this paper, anotheranti-dynamo theorem, called Cowling's theorem, which states that axisymmetricmagnetic fields cannot support dynamo action, is carefully tested against thicktubular and curved Riemannian untwisted flows, as well as thin flux tubes indiffusive and diffusionless media. In the non-diffusive media the Cowling'stheorem is not violated in thin Riemann-flat untwisted flux tubes, where theFrenet curvature is negative. Nevertheless the diffusion action in the thinflux tube leads to a a dynamo action driven by poloidal flows as shown by Loveand Gubbins (Geophysical Res.) in the context of geodynamos. Actually it isshown that a slow dynamo action is obtained. In this case the Frenet andRiemann curvature still vanishes. In the case of magnetic filaments indiffusive media dynamo action is obtained when the Frenet scalar curvature isnegative. Since the Riemann curvature tensor can be expressed in terms of theFrenet curvature of the magnetic flux tube axis, this result can be analogousto a recent result obtained by Chicone, Latushkin and Smith, which states thatgeodesic curvature in compact Riemannian manifolds can drive dynamo action inthe manifold. It is also shown that in absence of diffusion, magnetic energydoes not grow but magnetic toroidal magnetic field can be generated by thepoloidal field, what is called a plasma dynamo.

Dynamos driven by poloidal flows in untwisted, curved and flat Riemannian diffusive flux tubes