Extending $\mathbb R^3$ coordinate systems concepts

Question

I was thinking about how to use different coordinate systems in 3D space and how to describe curved surfaces embedded in 3D space when I realized that all the notations I know make sense only if everything is embedded $\mathbb R^3$, where I can use cartesian coordinates. In particular, what made me understand about this limitation is the following:

Given a certain coordinate system $(q_1,q_2...)$ a definition for tangent basis is $\vec e_{q_i}=\frac {\partial \vec P}{\partial {q_i}}$. Using this definition the derivative of a vector field along the coordinate $q_j$ become: $\frac {\partial \vec A}{\partial q_j}=\frac {\partial A^i}{\partial q_j} \vec e_{q_i}+A^i\frac {\partial \vec e_{q_i}}{\partial q_j}$.

What's the problem? the terms $\frac {\partial \vec P}{\partial {q_i}}$ and $\frac {\partial \vec e_{q_i}}{\partial q_j}$. Indeed for me these terms make sense only if the space is embedded is $\mathbb R^3$ because then I can give to $P$ and $\vec e_{q_i}$a cartesian form that allows me to perform the derivative. For example for polar coordinates on a plane:

$\vec P= r \cos(\theta)\vec i+r \sin(\theta)\vec j$ form which I obtain $\vec e_\theta=-r\sin(\theta) \vec i+r \cos(\theta) \vec j$.

If I would like to describe a curved space from inside I wouldn't know how to interpret these terms (for example space-time in general relativity or to do an easier example an inhabitant of flatland that lives on a curved surface).

Does this make sense for you or do you think these concepts can be used also for the description of a curved space from inside? If it is possible, can you give me an intuitive idea of the interpretation of those terms that I point out?

Answer 1

There are two different ways to do differential geometry: the 'extrinsic' view, and the 'intrinsic' view.

• The extrinsic view is what you just described: you set up an embedding of your manifold inside a copy of $\mathbb R^n$, and you refer your manifold's geometry to that ambient space.

• The intrinsic view is what you're asking about: studying the geometry of the space without any reference to geometrical concepts that lie outside that space.

As it happens, the intrinsic view is perfectly possible, but you do need additional structures in place for it. So, for example:

• Since you can no longer use the idea of distance provided by the ambient space, you need your manifold to come equipped with a metric.
• Since you can no longer use the 'invariant directions' of the cartesian basis vectors of your space, if you want to compare tangent vectors at different (or neighbouring) points, then you will need a way to 'transport' vectors from one place to the other, and for this you need your manifold to come equipped with a suitable connection (which can often be derived from the metric).

These concepts completely supersede the notions in your question (including in particular the derivatives of the basis vectors and their expressions in terms of an invariant cartesian basis), which are not available in intrinsic geometry.

For more details, see a good textbook on differential geometry. I really like the first volume of Spivak's course, but it is a fairly long book, and there's probably shorter introductions out there.