  While stretching of a polymer along a flat surface is hardly different from
the classical Pincus problem of pulling chain ends in free space, the role of
curved geometry in conformational statistics of the stretched chain is an
exciting open question. Here by means of the scaling analyses and computer
simulations we examine stretching of a fractal polymer chain around a disc in
2D (or a cylinder in 3D) of radius $R$. Surprisingly, we reveal that the
typical excursions of the polymer away from the surface scale as $\Delta \sim
R^{\beta}$, with the Kardar-Parisi-Zhang (KPZ) growth exponent $\beta=1/3$, for
any fractal dimension of the chain. Moreover, we find that the
curvature-induced correlation length of a fractal chain behaves as $S^* \sim
R^{1/z}$ with the KPZ dynamic exponent $z=3/2$, suggesting that the crossover
from flat to curved geometry of a stretched polymer corresponds to the
crossover from large to short time scales in the KPZ stochastic growth. Thus,
we argue that curvature of an underlying boundary furnishes universal KPZ-like
statistics to the stretched fractal paths, which further suggests numerous
connections with several branches of mathematical physics.
