It then turns to manifolds, which were what was left of Euclidean space after Messrs. Lie, Riemann et cie were done with it. Almost anything which can be described by continuous variables can be represented by a manifold, but it seems at first that we've paid for this generality by being unable to do anything further. There's enough structure in a manifold, however, that one can define curves, as a set of functions from a single parameter to the coordinates of the manifold. Taking the derivative with respect to this parameter gives us tangents to the curve, which are vectors. Fiber bundles, which prove to be so useful later in physics, and seem so mysterious, prove to be little more than giving each point on the manifold its own vector-space.
Tensors are functions from vectors and one-forms to the real numbers, linear in all their arguments.
Older works defined vectors, forms and tensors by the way their representations in particular coordinate systems changed when the coordinate system changed. To my mind, this is about as illuminating as saying that a length is what increases by a factor of twelve when going from feet to inches. The new approach focuses instead on the underlying geometrical object, and the coordinate-transformation rules are easily derived from the definitions. The metric tensor is introduced at the end of the chapter. This is a symmetric, invertible tensor taking two vectors into a real number, and we use it to define the inner product of two vectors, as well as a way of converting vectors and one-forms into each other.