This appendix is intended to explain some of the basic ideas in differential geometry which we have taken for granted in the main text. There exist a number of excellent modern introductory texts, from amongst which we might suggest the very readable book of Gallot, Hulin and Lafontaine [Ga-Hu-La].

Differentiable manifolds and maps.

Let M be a compact metric space. We call M a (d-dimensional) Ckmanifold, where k ≥ 1, if there exists an open cover {Uα} for M and homeomorphisms xα: Uα→Vα onto open sets Vα⊆ℝd such that each composition xα ∘(xβ)−1 is a Ck map (on neighborhoods of ℝd) whenever it is defined.

If a subset V⊆M is also a manifold, then we call it a submanifold of M. If M is a C∞ manifold which has, in addition, a C∞ group operation then it is called a Lie group (for example, the torus Td = ℝd/ℤd with the operation (x+ℤd, y+ℤd) ↦ x+y+ℤd).

Definition. The maps xα = (xα1,…, xαd) are called local co-ordinates (or charts) for M.