  We derive a reduced quasi-one-dimensional theory of geometrically frustrated
elastic ribbons. Expressed in terms of geometric properties alone, it applies
to ribbons over a wide range of scales, allowing the study of their elastic
equilibrium, as well as thermal fluctuations. We use the theory to account for
the twisted-to-helical transition of ribbons with spontaneous negative
curvature, and the effect of fluctuations on the corresponding critical
exponents. The persistence length of such ribbons changes non-monotonically
with the ribbon's width, dropping to zero at the transition. This and other
statistical properties qualitatively differ from those of non-frustrated
fluctuating filaments.
