Can we choose a metric on quotient spaces so that the quotient map does not increase distances? This is trivially true, when the metric have an upper bound. But the problem is there is no nature way of doing this.

Consider a sphere $S$ in $\mathbb{R}^3$. We identify two points on the sphere. Maybe we would like that for the quotient map $p$, the tangent map $p_t:T_x S\rightarrow T_{p(x)} (S/\sim)$ is an isometry at smooth points. However, this is not possible. In fact you can prove it using differential geometric arguments.

For example, see this paper. http://www.sciencedirect.com/science/article/pii/S0001870813000923

The singularity can not be classified to any of the three kinds.

Can we choose a metric on quotient spaces so that the quotient map does not increase distances? This is trivially true, when the metric have an upper bound. But the problem is there is no natural way of doing this.

Consider a sphere $S$ in $\mathbb{R}^3$. We identify two points on the sphere. Maybe we would like that for the quotient map $p$, the tangent map $p_t:T_x S\rightarrow T_{p(x)} (S/\sim)$ is an isometry at smooth points. However, this is not possible. In fact you can prove it using differential geometric arguments.

For example, see this paper. http://www.sciencedirect.com/science/article/pii/S0001870813000923

The singularity can not be classified to any of the three kinds.