The dual notion is that of conormal bundle. The notion also makes sense for some other contexts, e.g. for smooth algebraic varieties.
There is always an isomorphism
but it is not canonically given, hence in particular cannot it general be chosen naturally. Except if certain additional structure is given:
Let be a smooth compact -dimensional manifold without boundary, then the question of triviality of the normal bundle for an embedding for sufficiently large does not depend on the embedding. For this one uses the fact that any two such embeddings are regularly homotopic (this means the existence of a smooth homotopy which is immersion for every and which induces on the level of differentials a homotopy for the tangent bundles) and that any two regular homotopies are themselves homotopic through regular homotopies leaving end points fixed. Then one just uses the homotopy invariance of vector bundles. Thus, if admits an embedding into with a trivial normal bundle then one says that has a stably trivial normal bundle. In that case, if is the union of with a disjoint base point, then there is a homeomorphism where denotes the -fold (reduced) suspension of based spaces .