  Westudythebehaviourofisocurvatureperturbationsinnon-linearsigmamodels which
naturally emerge in supergravity and string inflationary scenarios. We focus on
the case of negatively curved field manifolds which can potentially lead to a
geometrical destabilisation of isocurvature modes. We find however that heavy
fields are stable when their effective mass is computed on the attractor
background solution. On the other hand, we show that ultra-light fields can
potentially suffer from a geometrical instability when the background
trajectory is a geodesic in field space. In this case a full understanding of
the system is likely to require the use of non-perturbative methods.
