  We demonstrate the necessity of a Poincar\'e type inequality for those metric
measure spaces that satisfy Cheeger's generalization of Rademacher's theorem
for all Lipschitz functions taking values in a Banach space with the
Radon-Nikodym property. This is done by showing the existence of a rich
structure of curve fragments that connect near by points, similar in nature to
Semmes's pencil of curves for the standard Poincar\'e inequality. Using
techniques similar to Cheeger-Kleiner, we show that our conditions are also
sufficient.
  We also develop another characterization of "RNP Lipschitz differentiability
spaces" by connecting points by curves that form a rich structure of partial
derivatives that were first discussed in work by the first author.
