Local Fourier transforms, analogous to the l-adic local Fourier
transforms [14], are constructed for connections over k((t)).
Following a program of Katz [12], a meromorphic connection on a curve is
shown to ber igid, i.e. determined by local data at the singularities,
if and only if a certain infinitesimal rigidity conditionis satisfied.
As in [12],the argument uses local Fourier transforms to prove an
invariance result for the rigidity index under global Fourier transform.
A key technical tool is the notion of good lattice pairs for a connection[5].