© 2015 American Physical Society. The Schwarzschild exterior space-time is conformally related to a direct product space-time, M2×S2, where M2 is a 2D 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 M2 that are useful for black hole physics. We prove that M2 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 M2 numerically and point out how these results will be used to establish global properties of Green functions on the Schwarzschild black hole space-time.