# How to determine whether the preimage of a point is a imbedding submanifold?

How to determine whether the preimage of a point is a imbedding submanifold??

Dear Folks:
It is well known that the preimage of a regular point is a imbedding submanifold, but is it possible that the preimage of a critical point is also a imbedding submanifold?? More generally, is there a prosedure to determine when the preimage of a point is a imbedding ??
Many thanks!!

quasar987
Homework Helper
Gold Member

It is possible that the preimage of a critical point is an embedded submanifold. For instance, consider the map f:R²-->R : (x,y) --> y² and let M:=graph(f)={(x,y,z) in R³ | z=f(x,y)=y²} This is the manifold that you get by taking a parabola parabolla in the yz plane and "sliding" it along the x axis so as to have one such parabola standing on each point (x,0,0). Now let h:M-->R be the height function h(x,y,z)=z. Then h-1(0)=R x {0} x {0} is an embedded submanifold, but each point of h-1(0) is critical because the derivative of h vanishes there.

mathwonk