One immediate benefit of considering coordinate-free descriptions of geometric objects is that we may talk about “curves” that are not a priori embedded in \mathbb{R}^3. In other words, we don’t have to start with a subset of \mathbb{R}^3 to be able to study 1-dimensional objects. There is already quite a nontrivial question we can ask: what curves can be embedded in a plane? The answer will be provided as a condition on k and \tau, and this description has the advantage of having nothing to do with \mathbb{R}^3 or any other non-intrinsic data. Later we will talk about surfaces (2-dimensional manifolds) that do not live in 3-space, but rather in 4-space. Having an intrinsic way of seeing objects is liberating and opens up new possibilities.
- See more at: http://blogs.ams.org/mathgradblog/#sthash.gsmEYMsj.dpuf
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