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Chicone et al [CMP (1995)] have shown that, kinematic fast dynamos indiffusive media, could exist only on a closed, 2D Riemannian manifold ofconstant negative curvature. This report, shows that their result cannot beextended to oscillatory ${\alpha}^{2}$-dynamos, when there are resonance modes,between toroidal and poloidal frequencies of twisted magnetic flux tubes. Thus,dynamo action can be supported in regions, where Riemannian curvature ispositive. For turbulent dynamos, this seems physically reasonable, sincerecently, [Shukurov et al PRE (2008)] have obtained a Moebius flow strip insodium liquid, torus Perm dynamo where curvature is also connected to themagnetic fields via diffusion. This could be done, by adjusting thecorresponding frequencies till they achieved resonance. Actually 2D torus, is amanifold of zero mean curvature, where regions of positive and negativecurvatures exist. It is shown that, Riemannian solitonic surface, endowed witha steady ${\alpha}^{2}$-dynamo from magnetic filamentary structures [Wilkin etal,PRL (2007)].

Resonant oscillations in ${\alpha}^{2}$-dynamos on a closed, twisted Riemannian 2D flux tubes