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I can totally understand how distance is naturally defined differently on a 2-dimensional sphere as opposed to ℝ^2. If you want to move from the point (x,y,z) to (x+dx,y+dy,z) in the plane, you have to move √(dx^2+dy^2) units, but if you want to make that same move in the x-y plane but on the surface of a sphere, you have no choice but to change your z coordinate too if you want to stay on the sphere, which adds a little bit of extra distance under the radical.

What I just summarized there is something that makes complete sense to me. I started with a fundamental concept of distance (the Euclidean metric) and then decided to twist the 2-dimensional subspace {(x,y,z)|z=0} of ℝ^3 into the 2-sphere. My fundamental concept of distance hasn't changed, but the actual distances that I will calculate

*naturally*change with the way that I changed the surface.

That's what I was hoping differential geometry would be: you define some manifold (a "smooth" subset of a higher-dimensional Euclidean space) and then just see what distances

*naturally*turn out to be using basic calculus. But apparently I'm wrong in some way.

Here is a list of concepts that I think I understand, which will be followed by my current and hopefully false understanding of what begets what:

-Metric tensor

-Topology (not the discipline, but the collection τ of subsets of a set X)

-Topological space

-Metric space

-Manifold

-"Shape" of a manifold (something I made up, probably)

Does topology induce a metric, or vice versa, or are they independent things? Does the topology even

*do*anything or does it just sit there smiling because it satisfies 4 rules? I feel like the intuitive notion of the "shape" of a space should naturally give rise to a distance function which is fundamentally based on Euclidean distances, but the metric coefficients are in general not equal to the Kronecker Delta to account for the curving and stretching--the "shape"--of the space. Does the "shape" of the space depend on the topology, or just the set of points X?

So to summarize, I would have thought that "shape" of the manifold is determined by exactly what smooth subset of a higher-dimensional Euclidean space you're sampling from, and that shape then determines the distance function at every point which only alters the Euclidean (or Lorentzian) distance for the purpose of correcting for curvature. But it seems like what happens instead is that you just define a distance function at every point, which then induces the "shape" of the manifold, which allows you to define a manifold without embedding it. Maybe it's mathematically equivalent, but it seems backwards to me. Furthermore, what's so evil about embedding?

This is a bit of a ramble, so I want to end with a more succinct question if everything I've written is so wrong you don't know where to begin in correcting me: what should I learn first? I know that I don't rigorously understand most of what I'm talking about, but I have to start somewhere, and I want to start at the thing from which everything else branches off. Is that topology, or differential geometry, or what?