It is hard to imagine a physical problem which does not involve some sort of continuous space. It might be physical three-dimensional space, four-dimensional spacetime, phase space for a problem in classical or quantum mechanics, the space of all thermodynamic equilibrium states, or some still more abstract space. All these spaces have different geometrical properties, but they all share something in common, something which has to do with their being continuous spaces rather than, say, lattices of discrete points. The key to differential geometry's importance to modern physics is that it studies precisely those properties common to all such spaces. The most basic of these properties go into the definition of the differentiable manifold, which is the mathematically precise substitute for the word ‘space’.

Definition of a manifold

As in §1.1, we denote by Rn the set of all n-tuples of real numbers (x1, x2,…, xn). A set (of ‘points’) M is defined to be a manifold if each point of M has an open neighborhood which has a continuous 1-1 map onto an open set of Rn for some n. (The reader unsure of what a 1-1 map onto something means should look at § 1.2.) This simply means that M is locally ‘like’ Rn. The dimension of M is, of course, n.