  The product spacetimes of constant curvature describe in Einstein gravity,
which is linear in Riemann curvature, Nariai metric which is a solution of
$\Lambda$-vacuum when curvatures are equal, $k_1=k_2$, while it is
Bertotti-Robinson metric describing uniform electric field when curvatures are
equal and opposite, $k_1=-k_2$. We probe pure Lovelock gravity by these simple
product spacetimes and prove that the same characterization of these solutions
is indeed true in general for pure Lovelock gravitational equation of order $N$
in $d=2N+2$ dimension. We also consider these solutions for the conventional
setting of Einstein-Gauss-Bonnet gravity.
