Contents

Part IGeneral methods2Differential geometry without a metric2.1 IntroductionThe concept of a tensor is often based on the law of transformation ofthe components under coordinate transformations, so that coordinatesare explicitly used from the beginning. This calculus provides adequatemethods for many situations, but other techniques are sometimes moreeffective. In the modern literature on exact solutions coordinatefree geometricconcepts, such as forms and exterior differentiation, are frequentlyused: the underlying mathematical structure often becomes more evidentwhen expressed in coordinatefree terms.Hence this chapter will present a brief survey of some of the basic ideasof differential geometry. Most of these are independent of the introductionof a metric, although, of course, this is of fundamental importance in thespace-times of general relativity; the discussion of manifolds with metricswill therefore be deferred until the next chapter. Here we shall introducevectors, tensors of arbitrary rank, p-forms, exterior differentiation and Liedifferentiation, all of which follow naturally from the definition of a differentiablemanifold. We then consider an additional structure, a covariantderivative, and its associated curvature; even this does not necessarily involvea metric. The absence of any metric will, however, mean that it willnot be possible to convert 1-forms to vectors, or vice versa.9