The structural connectome is often represented by fiber bundles generated from various types of tractography. We propose a method of analyzing connectomes by representing them as a Riemannian metric, thereby viewing them as points in an infinite-dimensional manifold. After equipping this space with a natural metric structure, the Ebin metric, we apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. We demonstrate connectome registration and atlas formation using connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project.