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Is there a criterion to show that a level set of some map is not an (embedded) submanifold? In particular, an exercise in Lee's smooth manifolds book asks to show that the sets defined by $x^3 - y^2 = 0$ and $x^2 - y^2 = 0$ are not embedded submanifolds.

In general, is it possible that a level set of a map which does not has constant rank on the set still defines a embedded submanifold?

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    Yes it is possible, e.g. $x^3-y^3=0$2011-05-03
  • 0
    Note that the constant rank theorem requires the rank to be constant *near* the level set and not only *on* it.2011-05-04

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