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

Shape and Fluctuations of Positively Curved Ribbons