Let be a smooth map between manifolds. We say that a point is a “regular point” if the derivative has rank ; otherwise, we say that is a “critical point”. A point is called a “regular value” if its preimage contains no critical points.
The first thing to notice is that this is only nontrivial if . If then can have rank at most , and thus every point is critical. Another observation is that is then is automatically regular; if its preimage is empty then it cannot contain any critical points.
Regular values are useful because of the generalization of the first part of the implicit function theorem: if is a regular value of , then is a topological manifold of dimension . Or, to put it another way, is a submanifold of “codimension” . Further, there is a unique differentiable structure for which is a smooth submanifold of .
Indeed, let be a coordinate patch around with . Given , pick a coordinate patch of with . Let be the projection onto the first components; let be the projection onto the last components; an let be the inclusion of the subspace whose first components are .
Now, we can write down the composition . Since this has (by assumption) maximal rank at , the implicit function theorem tells us that there is a coordinate patch in a neighborhood of such that . So we can set , which is open in , and get
Setting we conclude that , since all these points are sent by to the preimage .
Now we claim that is not just any subset of , but in fact . Clearly is contained in this intersection, since
On the other hand, if is in this intersection, then for a unique — unique because and are both coordinate maps and thus invertible — and we have
meaning that the first components of must be , and thus . Thus .
Therefore maps homeomorphically onto a neighborhood of in the subspace topology induced by . But this means that acts as a coordinate patch on ! Since every point can be found in some local coordinate patch, is a topological manifold. For its differentiable structure we’ll just take the one induced by these patches.
Finally, we have to check that the inclusion is smooth, so is a smooth submanifold — that its differentiable structure is compatible with that of . But this is easy, since at any point we can go through the above process and get all these functions. We check smoothness by using local coordinates on and on , concluding that , which is clearly smooth.