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Given strong local Dirichlet forms and $\mathbb{R}^N$-valued functions on ametrizable space, we introduce the concepts of geodesic distance and intrinsicdistance on the basis of these objects. They are defined in a geometric and ananalytic way, respectively, and they are closely related with each other insome classical situations. In this paper, we study the relations of thesedistances when the underlying space has a fractal structure. In particular, weprove their coincidence for a class of self-similar fractals.

Geodesic distances and intrinsic distances on some fractal sets