  The aim of this paper is to establish two fundamental measure-metric
properties of particular random geometric graphs. We consider
$\varepsilon$-neighborhood graphs whose vertices are drawn independently and
identically distributed from a common distribution defined on a regular
submanifold of $\mathbb{R}^K$. We show that a volume doubling condition (VD)
and local Poincar\'e inequality (LPI) hold for the random geometric graph (with
high probability, and uniformly over all shortest path distance balls in a
certain radius range) under suitable regularity conditions of the underlying
submanifold and the sampling distribution.
