Submanifolds vs Manifolds / Immersion vs Charts

  • Thread starter Trifis
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Ok first of all I'd like to mention that I've searched the forum and didn't find anything similar, so hopefully this thread is not unwelcome...

Now as the title suggests, I am interested in a parallelization of the two concepts. Personally I like to introduce the idea of submanifolds prior to that of manifolds, because they can be identified with the geometrical objects (curves, surfaces) handled in regular multivariable calculus. Consequently, the immersion may be described as the usual parametrization of these objects up to the condition about open sets mapping (later defined as homeomorphism in topology).
But manifolds take a different approach, since they require charts (homeomorphic maps from a space of some dimension to the euclidean space of the same dimension) for their definition. Thus, while immersions map points of a coordinate space to the submanifold, the charts operate in the inverse way and project points of a general space onto an euclidean equivalent.

For example: let us consider the 2-sphere. It can be defined as a 2-dimensional submanifold M of codimension 1 by the use of the immersion (=parametrization) φ: U[itex]\subset[/itex]ℝ2→(M[itex]\cap[/itex]V)[itex]\subset[/itex]ℝ3 , (θ,φ)→(x,y,z) , with U and V open sets and φ a hemeomorphism.
Alternatively, the sphere S2[itex]\subset[/itex]ℝ3 with the relative topology is also a topological manifold. Should we make use of the stereographic projection, we acquire two maps (fi: U[itex]\subset[/itex]S2→Vi[itex]\subset[/itex]ℝ3[itex]\subset[/itex]ℝ2 , i=1,2) with differentiable transition maps, which helps us make the sphere this time a differentiable manifold!

Is my train of thought valid so far? Does sth need to be presented more rigorously?
 

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  • #2
HallsofIvy
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Certainly low dimensional manifolds are easier to think of than large dimensional manifolds but I can see no reason for introducing "sub-manifold" before "manifold". How can you have a "sub-object" before you said what the "object" is?
 
  • #3
quasar987
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Certainly low dimensional manifolds are easier to think of than large dimensional manifolds but I can see no reason for introducing "sub-manifold" before "manifold". How can you have a "sub-object" before you said what the "object" is?
I think he is refering to "submanifold of R^n". The idea is to treat these objects as a motivation for the definition of abstract manifolds and their tangent spaces. It is the route employed by the authors Gallot-Lafontaine-Hullin, and also Spivak.

to the OP: I advise you to use the terminology "parametrization" instead of immersion because an immersion is a much more general map btw manifolds.
 
  • #4
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I think he is refering to "submanifold of R^n". The idea is to treat these objects as a motivation for the definition of abstract manifolds and their tangent spaces. It is the route employed by the authors Gallot-Lafontaine-Hullin, and also Spivak.
Exactly that is the concept. Submanifolds of ℝn give a first idea about spaces that may be not linear (not even vector spaces) but they interract with linear spaces through maps.
@HallsofIvy have a look at my trivial example and how do I infer the general from the particular.
to the OP: I advise you to use the terminology "parametrization" instead of immersion because an immersion is a much more general map btw manifolds.
Yes of course, it is more general, but for an introduction to the topic, isn't it right to think of it as a parametrization and its inverse as the chart ?
 
  • #5
lavinia
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The idea of manifold and a submanifold of Euclidean space are technically different since a manifold is defined through an intrinsic topology composed of coordinate charts while a submanifold is defined through the extrinsic topology of Euclidean space.

It is a theorem that any manifold can be realized as a submanifold of Euclidean space but a priori this is not clear.

While parameterized surfaces probably were the first studied manifolds, I would guess that the modern definition of manifold comes from notions of intrinsic geometry discovered by Gauss and Riemann and maybe also from the idea of a Riemann surface which does not come from parameterized surfaces but from finding the natural domain of definition of analytic functions.
 
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