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**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!!