  Random geometric graphs result from taking $n$ uniformly distributed points
in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean
distance is at most $r$, for some prescribed $r$. We show that monotone
properties for this class of graphs have sharp thresholds by reducing the
problem to bounding the bottleneck matching on two sets of $n$ points
distributed uniformly in $[0,1]^d$. We present upper bounds on the threshold
width, and show that our bound is sharp for $d=1$ and at most a sublogarithmic
factor away for $d\ge2$. Interestingly, the threshold width is much sharper for
random geometric graphs than for Bernoulli random graphs. Further, a random
geometric graph is shown to be a subgraph, with high probability, of another
independently drawn random geometric graph with a slightly larger radius; this
property is shown to have no analogue for Bernoulli random graphs.
