  We study the normal approximation of functionals of Poisson measures having
the form of a finite sum of multiple integrals. When the integrands are
nonnegative, our results yield necessary and sufficient conditions for central
limit theorems. These conditions can always be expressed in terms of
contraction operators or, equivalently, fourth cumulants. Our findings are
specifically tailored to deal with the normal approximation of the geometric
$U$-statistics introduced by Reitzner and Schulte (2011). In particular, we
shall provide a new analytic characterization of geometric random graphs whose
edge-counting statistics exhibit asymptotic Gaussian fluctuations, and describe
a new form of Poisson convergence for stationary random graphs with sparse
connections. In a companion paper, the above analysis is extended to general
$U$-statistics of marked point processes with possibly rescaled kernels.
