In both of our pathological examples last time, the problems were very isolated. They depended on two separated parts of the domain manifold interacting with each other. And since manifolds can be carved up easily, we can always localize and find patches of the domain where the immersion map is a well-behaved embedding.
More specifically, if is an immersion, with always an injection for every , then for every point there exists a neighborhood of and a coordinate map around so that if and only if . Further, the restriction is an embedding.
This is basically the actual extension of the second part of the implicit function theorem to manifolds. Appropriately, then, we’ll let be the same inclusion into the first coordinates. We pick a coordinate map around with , and another map around with . Then we get a map from a neighborhood of to a neighborhood of .
Now, the assumption on is that is injective, meaning it has maximal rank at every point. Since and are diffeomorphisms, the composite also has maximal rank at . The implicit function theorem tells us there is a coordinate map in some neighborhood of and a neighborhood of such that .
We set , and , restricting the domain of , if necessary. This establishes the first part of our assertion. Next we need to show that is an embedding. But , which is a composition of embeddings, and is thus an embedding itself.
If is already an embedding at the outset, then for some open . In this case, with as in the theorem, we have
That is, there is always a set of local coordinates in so that the image of is locally the hyperplane spanned by the first of them.