Showing that a level set is not a submanifold 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? 

The set given by $(x^2+y^2)(x^2+y^2-1)=0$ is an embedded submanifold in $\mathbb R^2$
 but it has components of different dimension and so I guess the map does not have constant rank on the set, but I haven't checked.
