  Poisson processes in the space of $k$-dimensional totally geodesic subspaces
($k$-flats) in a $d$-dimensional standard space of constant curvature
$\kappa\in\{-1,0,1\}$ are studied, whose distributions are invariant under the
isometries of the space. We consider the intersection processes of order $m$
together with their $(d-m(d-k))$-dimensional Hausdorff measure within a
geodesic ball of radius $r$. Asymptotic normality for fixed $r$ is shown as the
intensity of the underlying Poisson process tends to infinity for all $m$
satisfying $d-m(d-k)\geq 0$. For $\kappa\in\{-1,0\}$ the problem is also
approached in the set-up where the intensity is fixed and $r$ tends to
infinity. Again, if $2k\le d+1$ a central limit theorem is shown for all
possible values of $m$. However, while for $\kappa=0$ asymptotic normality
still holds if $2k>d+1$, we prove for $\kappa=-1$ convergence to a non-Gaussian
infinitely divisible limit distribution in the special case $m=1$. The proof of
asymptotic normality is based on the analysis of variances and general bounds
available from the Malliavin--Stein method. We also show for general
$\kappa\in\{-1,0,1\}$ that, roughly speaking, the variances within a general
observation window $W$ are maximal if and only if $W$ is a geodesic ball having
the same volume as $W$. Along the way we derive a new integral-geometric
formula of Blaschke--Petkantschin type in a standard space of constant
curvature.
