INTRODUCTION

Stretching of line elements in fluids with or without conductivity is studied from various points of view. Firstly, simultaneous concentration of vortex and magnetic tubes is considered in section 2 by presenting an exact solution of the axisymmetric MHD equation for a viscous, incompressible, conducting fluid. This solution (Kambe 1985) tends to a stationary state that results from complete balance of convection, diffusion and stretching. Sections 3 and 4 are concerned with mathematical formulation based on Riemannian differential geometry, and global (mean) stretching of line elements is considered.

The general form of the Riemannian curvature tensor for any Lie group was derived by Arnold (1966), where explicit formulae for T2 (two-torus) were given. Explicit expressions for diffeomorphism curvatures on Tn (and even on any locally flat manifold) were described by Lukatskii (1981). Recently, Nakamura et al. (1992) considered the curvature form on T3 corresponding to three-dimensional motion of an ideal fluid with periodic boundary conditions in a cubic space. An immediate consequence is the property that the section curvature in the space of ABC diffeomorphisms is a negative constant for all the sections (Kambe et al. 1992).