We define sets with finitely ramified cell structure, which are
generalizations of p.c.f. self-similar sets introduced by Kigami and of
fractafolds introduced by Strichartz. In general, we do not assume even local
self-similarity, and allow countably many cells connected at each junction
point. We prove that if Kigami's resistance form satisfies certain assumptions,
then there exists a weak Riemannian metric such that the energy can be
expressed as the integral of the norm squared of a weak gradient with respect
to an energy measure. Furthermore, we prove that if such a set can be
homeomorphically represented in harmonic coordinates, then for smooth functions
the weak gradient can be replaced by the usual gradient. We also prove a simple
formula for the energy measure Laplacian in harmonic coordinates.