For much of physics and mathematics the concept of a continuous map, provided by topology, is not sufficient. What is often required is a notion of differentiable or smooth maps between spaces. For this, our spaces will need a structure something like that of a surface in Euclidean space ℝn. The key ingredient is the concept of a differentiable manifold, which can be thought of as topological space that is ‘locally Euclidean’ at every point. Differential geometry is the area of mathematics dealing with these structures. Of the many excellent books on the subject, the reader is referred in particular to [1–14].

Think of the surface of the Earth. Since it is a sphere, it is neither metrically nor topologically identical with the Euclidean plane ℝ2. A typical atlas of the world consists of separate pages called charts, each representing different regions of the Earth. This representation is not metrically correct since the curved surface of the Earth must be flattened out to conform with a sheet of paper, but it is at least smoothly continuous. Each chart has regions where it connects with other charts – a part of France may find itself on a map of Germany, for example – and the correspondence between the charts in the overlapping regions should be continuous and smooth. Some charts may even find themselves entirely inside others; for example, a map of Italy will reappear on a separate page devoted entirely to Europe.