  The Schwarzschild exterior space-time is conformally related to a direct
product space-time, $\mathcal{M}_2 \times S_2$, where $\mathcal{M}_2$ is a
two-dimensional space-time. This direct product structure arises naturally when
considering the wave equation on the Schwarzschild background. Motivated by
this, we establish some geometrical results relating to $\mathcal{M}_2$ that
are useful for black hole physics. We prove that $\mathcal{M}_2$ has the rare
property of being a causal domain. Consequently, Synge's world function and the
Hadamard form for the Green function on this space-time are well-defined
globally. We calculate the world function and the van Vleck determinant on
$\mathcal{M}_2$ numerically and point out how these results will be used to
establish global properties of Green functions on the Schwarzschild black hole
space-time.
