Physics usually begins with some concept of a space of events, or of positions at which objects or fields can be present. While this could be a discrete space (e.g. in lattice models) or a topological space without extra structure, it is usually assumed to be a continuum on which one can carry out the operations of calculus, i.e. a differential manifold.
Most of modern theoretical physics can be written in the language of differential geometry and topology, though it has only become common to do so since gauge theories assumed their present prominent role. Many advanced notions in these areas find a place in physics (see e.g. Nakahara (1990)). We shall be careful below to distinguish between concepts dependent on and independent of the presence of a metric, since gauge theories usually do not assume a metric.
While it is not true that every geometric object is of physical significance, it will be true of the geometric quantities we discuss in subsequent chapters. So when we consider geometric questions, it is important to recognize that these can also be understood as physical. Indeed, one of our aims is to show how powerful geometric methods can be in discussing cosmological questions, once one has mastered the necessary tools.
Dirac in his classic book (Dirac, 1981) emphasizes that quantum mechanics rests on a number of principles. The first two are that observables are operators, which can be expressed in different bases, and that the core of the dynamics lies in non-zero commutators for these operators.