In CMU DDG course Lecture 12 “Smooth Surfaces I”, the “abstractness” of the Riemannian metric means as long as an inner product is assigned to each point in the geometric object, it’s enough for us to compute quantities such as distances, angles, lengths, areas, etc. formed by vectors represented in the local coordinate chart, without the need to embed the geometric object into some higher dimensional space for visualization.

This is a very important concept and methodology in that we only perform abstract computations based on Riemannian metric, in order to know, characterize, transform and control the properties of the geometric object without actually seeing or feeling it. This again embodies the power of mathematical abstraction. This is also a counter-example of the popular pedagogy, which introduces lots of visualizations, animations and jokes into classes just to make the courses appealing to students. However, such “obsequious” style is weak and limited. I’m in favour of the abstract style.