- #1

Trifis

- 167

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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]ℝ

^{2}→(M[itex]\cap[/itex]V)[itex]\subset[/itex]ℝ

^{3}, (θ,φ)→(x,y,z) , with U and V open sets and φ a hemeomorphism.

Alternatively, the sphere S

^{2}[itex]\subset[/itex]ℝ

^{3}with the relative topology is also a topological manifold. Should we make use of the stereographic projection, we acquire two maps (f

_{i}: U[itex]\subset[/itex]S

^{2}→V

_{i}[itex]\subset[/itex]ℝ

^{3}[itex]\subset[/itex]ℝ

^{2}, 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?