In this chapter we introduce two additional structures on a differentiable manifold. First we consider the “parallel transport” of a vector, which allows a vector at one point on the manifold to be “transported” along a path to another point, where it can then be compared to other vectors at the new point. This idea gives rise to a number of interdependent quantities, and is particularly important in physics, where it is generalized to gauge theories.

We then consider the introduction of a metric, an inner product in each tangent space that permits us to define lengths of vectors and angles between them. A metric determines a unique associated parallel transport, and is the fundamental quantity in general relativity. We then touch upon some other structures on manifolds that appear in physics.