N-sphere as manifold without embedding

Twigg

Gold Member
Also, could you please give me an example of a manifold where not embedding it simplifies things? I am trying to remove this thought that I have that a manifold must be embedded in Euclidean space in order to be useful/interesting to us. I know that my point of view is wrong (based on more knowledgeable people telling me so), but I need to be more convinced.
The real projective plane $\mathbb{RP}^2$ comes to mind. It has no embeddings in $\mathbb{R}^3$ but it's most easily visualized using surfaces in $\mathbb{R}^3$ as visual aids, like the crosscap. You can even visualize it with a 2D figure if you count its fundamental rectangle. Those methods are much more useful than trying to visualize it as an embedding in $\mathbb{R}^4$. The formal definition doesn't mention embeddings either, it's just a quotient space of the sphere with a smooth structure. The reason embeddings are useful and are used to characterize manifolds a lot is because they're a common, convenient starting point for constructing manifolds.

The study of smooth manifolds without geometry is called Differential Topology. I do not know Lee's book but a short and wonderful introduction is Milnor's Topology from the Differentiable Viewpoint. The last chapter discusses the classification of smooth mappings of manifolds into spheres.
The first volume of Spivak's Differential Geometry is another great resource. It dedicates a few solid chapters to differential topology, dedicating a chapter to each layer of structure in the manifold cake. Spivak was Milnor's student and those first few chapters are a good introduction to Milnor's work on exotic spheres, which helps puts a lot that's been said in this thread in perspective. Strongly recommend to JonnyG and anyone interested.

zinq

Just to agree with Twigg: If we have a manifold with a nice embedding in Euclidean space given, it is very rare that this ever "gets in the way".

Often having a concrete representation of a manifold as a subset of Euclidean space is actually handy for manipulation — even when the Riemannian metric inherited from Euclidean space may not be the one we're interested in (if we are interested in any Riemannian metric at all).

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