So this is the definition I have:
Given a smooth map of differential manifolds, a point in the image manifold is termed a regular value for the smooth map if every point in its inverse image is a regular point, i.e. if the map from the tangent space at any point in the inverse image, is surjective.
This is how I am understanding things.
So taking it part by part, we have a map from one manifold to another. So far so good.
Because this map need not be bijective, multiple points in the domain may map to the image set.
A regular value is one such that, if you are given a point in the target manifold, and all the points that map to it in the domain manifold, then for every tangent plane to each of the points in the domain, there is a bijection from the tangent plane at the domain point to the tangent plane at the target point.
But. isn't that always true? Aren;t 2 planes always trivially bijective? What am I missing?
Best Answer
No that isn't always the case. For example the inclusion $\mathbb{R} \to \mathbb{R}^2, t \mapsto (t, 0)$ isn't regular on its image, since the tangent spaces of $\mathbb{R}$ are one-dimensional while those of $\mathbb{R}^2$ are two dimensional. In general, for $n < m$ a map from a $n$-manifold to a $m$-manifold isn't regular on its image.
The situation above isn't the only reason why a point can fail to be regular though. The map $\mathbb{R}^2 \to \mathbb{R}^2$ given by $(x, y) \mapsto (x, xy)$ isn't regular at $(0, 0)$.