Identifying Elements of a Manifold without Explicit Embedding in Higher Dimensional Spaces

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.

• 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 – Diesirae92 Dec 28 '17 at 20:16