  Given an increasing graph property $\mathcal{P}$, a graph $G$ is
$\alpha$-resilient with respect to $\mathcal{P}$ if, for every spanning
subgraph $H\subseteq G$ where each vertex keeps more than a
$(1-\alpha)$-proportion of its neighbours, $H$ has property $\mathcal{P}$. We
study the above notion of local resilience with $G$ being a random geometric
graph $G_d(n,r)$ obtained by embedding $n$ vertices independently and uniformly
at random in $[0,1]^d$, and connecting two vertices by an edge if the distance
between them is at most $r$.
  First, we focus on connectivity. We show that, for every $\varepsilon>0$, for
$r$ a constant factor above the sharp threshold for connectivity $r_c$ of
$G_d(n,r)$, the random geometric graph is $(1/2-\varepsilon)$-resilient for the
property of being $k$-connected, with $k$ of the same order as the expected
degree. However, contrary to binomial random graphs, for sufficiently small
$\varepsilon>0$, connectivity is not born $(1/2-\varepsilon)$-resilient in
$2$-dimensional random geometric graphs.
  Second, we study local resilience with respect to the property of containing
long cycles. We show that, for $r$ a constant factor above $r_c$, $G_d(n,r)$ is
$(1/2-\varepsilon)$-resilient with respect to containing cycles of all lengths
between constant and $2n/3$. Proving $(1/2-\varepsilon)$-resilience for
Hamiltonicity remains elusive with our techniques. Nevertheless, we show that
$G_d(n,r)$ is $\alpha$-resilient with respect to Hamiltonicity for a fixed
constant $\alpha = \alpha(d)<1/2$.
