[Currently reading Geometry of Surfaces (Stillwell)]
One can create a Torus by joining opposite sites of a rectangle in $\mathbb{R}^2$. When joining the second pair, the lengths from $\mathbb{R}^2$ get distorted because of the circular form of the torus.
The book now says, one can define the torus as a quotient space $\mathbb{R}^2 / \Gamma$ where $\Gamma$ is the group of transformations , generated by the two transformations which moves the respectively opposite sites together. Thus, one can carry the local geometry of $\mathbb{R}^2$ over to $\mathbb{R}^2 / \Gamma$.
Sure, this does resolve the problem of length distortion, but why would someone conduct euclidean geometry with the "wrong" lengths ?