Book analysis on metric spaces and geodesics.

Question

I am looking for a book that defines and proves results about absolutely continuous curves in metric (i.e. curves of the form $\omega :[a,b]\to X$ where $X$ is a metric space) spaces and geodesic in metric spaces, in particular, things like

The following facts are equivalent

1. $\omega$ is a constant-speed geodesic,
2. $\omega\in\operatorname{AC}(X)$ and $|\omega'|(t)=d(\omega(0),\omega(1))$ a.e
3. $\omega$ solves $\min\{\int_0^1|\omega'|(t)^pdt:\omega(0)=x_0,\omega(1)=x_1\}$

Is there any good reference?