Geometry of Large Extra Dimensions versus Graviton Emission

We study how the geometry of the large extra dimensions may affect field theory results on a three-brane. More specifically, we compare cross sections for graviton emission from a brane when the internal space is a N-torus and a N-sphere for N=2 to 6. The method we present can be used for other smooth compact geometries. We find that the ability of high energy colliders to determine the geometry of the extra dimensions is limited but there is an enhancement when both the quantum gravity scale and N are large. Our field theory results are compared with the low energy corrections to the gravitational inverse square law due to large dimensions compactified on other spaces such as Calabi-Yau manifolds.