In most introductions to manifolds, texts typically describe how most n-dimensional manifolds can either described extrinsically, as surfaces embedded in a higher dimensional space, or intrinsically as a topological space equipped with charts.

This is all fairly elementary. However, if I want to learn to think about curvature and other properties as intrinsic to a manifold, I seem to need to be able to define any manifold without reference to extra dimensions. I'm having trouble finding a general means for doing this given a level set description of a manifold.

Take, for example, the unit sphere $\Bbb S^N$ in N dimensions. We can easily define it as a curve $x_1^2 + x_2^2 + ... x_{N+1}^2 = 1$ in $\Bbb R^{N+1}$, but it takes some concerted effort to properly find a definition of the space that doesn't make explicit reference to $\Bbb R^{N+1}$. Doing this for any arbitrary manifold seems even harder, but seems necessarily possible.

As so far, I see two potential solutions to this problem.

The first, and perhaps more difficult, is to find a general program for taking any n-dimensional manifold defined as a level set in a higher dimensional surface, and reconstructing the space from $\Bbb R^{N}$. In the unit sphere example, I've seen a quotient space description.

The second is more of conceptual gerrymandering on my part but considers the possibility that I've misunderstood the definition of a topological embedding. More precisely, when we embed something in a higher dimensional space, it implies we're mapping from some well defined set to other well defined (level?) subset of $\Bbb R^{M}$. Perhaps the $\Bbb R^{3}$ embedding of $\Bbb S^{2}$, or the level set embedding of a surface more generally, isn't the higher dimensional embedding that Whitney Embedding guarantees is possible. My sense of Whitney Embedding comes from Lee Smooth Manifolds Ch 6, but I may be misunderstanding him.

  • $\begingroup$ once you endow the manifold with a riemannian metric everything is fine. Just google Levi Civita connection and Ricci tensor (along with sectional cuurvature) and everything's gonna be clear $\endgroup$ – Diesirae92 Dec 28 '17 at 20:16

I think you're getting hung up on something that's not important. A manifold is something that obeys certain rules; there are lots of ways to construct manifolds; and there's no need for every manifold you encounter to be constructed by the same process. Constructing an abstract manifold by taking a subset of higher dimensional space and forgetting the embedding is a perfectly valid thing to do, and desirable in many situations.

You can also construct manifolds by gluing together open balls (formally, by taking a quotient of a disjoint union of balls). Indeed, many useful manifolds naturally arise in this manner and this is a big reason the intrinsic definition of a manifold is useful. If you've seen a surface covered by a triangular mesh, just imagine the same idea in arbitrary dimensions.

  • $\begingroup$ Your second point seems to be what I'm looking for. I think my difficulty came from this chicken-egg problem where I have some set and topological space and later on in life find charts that describe its dimensionality. However, this presented conceptual problems when considering that the notion of the dimensionality of the subset before we assigned it to charts seemed poorly defined. How do I know my sphere should map to R2 and not R3? $\endgroup$ – Dragonsheep Dec 28 '17 at 20:44
  • $\begingroup$ Only by constructing the sphere from R2 could I be sure. However, if I reverse the way I think about this and use the charts to construct the space, there’s no ambiguity. Of course, these are just conceptual tricks and nothing formal. $\endgroup$ – Dragonsheep Dec 28 '17 at 20:46
  • $\begingroup$ So it's worth distinguishing between a definition and a recipe for constructing something. The definition of a manifold is something that lets you check if something you already have is a manifold or not; it doesn't need to provide a way of constructing manifolds, a way of enumerating all manifolds that exist, or even a guarantee that there is such a thing as a manifold. (Compare how you can easily define an "odd perfect number," but there may well not be any.) $\endgroup$ – Daniel McLaury Dec 30 '17 at 17:28

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