Much of this chapter is geometric propaganda. It is intended to convince the reader that there is a ‘category’ of 6-Riemann manifolds in which one can work systematically. This 6-geometry can also be thought of as the geometry of manifolds with asymptotically cylindrical ends. There are other geometries which are similar to 6-geometry (see [55], [54], [56], [32], [31] and [63] for a general discussion).