  We study the shape and shape fluctuations of positively curved ribbons, with
a flat reference metric and a sphere-like reference curvature. Such
incompatible geometry is likely to occur in many self assembled materials and
other experimental systems. Such ribbons exhibit a sharp transition between
rigid ring and an anomalously soft spring as a function of their width. As a
result, the temperature dependence of these ribbons' shape is unique,
exhibiting a non-monotonic dependence of the persistence and Kuhn lengths on
the temperature and width. We map the possible configuration phase space and
show the existence of three phases- at high temperatures it is the Ideal Chain
phase, where the ribbon is well described by classical models (e.g- worm like
chain model); The second phase, for cold and narrow ribbons, is the Plain
Ergodic phase - a ribbon in this phase might be thought of as made out of
segments that gyrate within an oblate spheroid with extreme aspect ratio; The
third phase, for cold, wide ribbons, is a direct result of the residual stress
caused by the incompatibility, called Random Structured phase. A ribbon in this
phase behaves on large scales as an Ideal Chain, however the segments of this
chain are not straight, rather they may have different shapes, mainly helices
(both left and right handed) of various pitches.
