(adsbygoogle = window.adsbygoogle || []).push({}); "standard" Square is not a Smooth Submanifold of R^2

Hi, everyone:

I am trying to show the standard square in R^2, i.e., the figure made of the line

segments joining the vertices {(0,0),(0,1),(1,0),(1,1)} is not a submanifold of R^2.

Only idea I think would work here is using the fact that we can immerse (using inclusion)

the tangent space of a submerged manifold S into that of the ambient manifold M

, so that, at every p in S, T_pS is a subspace of T_pM .

Then the problem would be clearly at the vertices. I think we can choose a tangent

vectorX_p at, say, T_(0,0) S, and show that X_p cannot be identified with

a tangent vector in T_(0,0) R^2.

Seems promising, but it has not yet been rigorized for your protection.

Any ideas for making this statement more rigorous.?

Thanks.

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# Standard Square is not a Smooth Submanifold of R^2

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