Tensor Calculus and Riemannian Geometry

The attentive reader probably noticed that the concept of a Riemann metric on an open subset of ℝ n which we introduced in the last chapter, and the related material on geodesics, beg for a generalization to manifolds. Although this is not quite as straightforward as one might imagine, the work we have done in the last chapter gets us well underway. A serious problem we must address, of course, is how the concepts we introduced behave under change of coordinates. It turns out that what we have said on covariance and contravariance in manifolds is not quite enough: we need to discuss the notation of tensor calculus.