the constant-rank theorem
Suppose a smooth map f: M -> N has a differential whose rank — the number of independent directions it genuinely moves — is the same constant k at every point near p. The constant-rank theorem says that despite f possibly looking complicated, there are coordinates near p and near f(p) in which f does the dumbest possible thing of rank k: it keeps the first k coordinates and zeroes out the rest. It is the structural backbone behind immersions, submersions, and the level-set theorems.
Precisely, if df has constant rank k on a neighborhood of p, then there exist a chart centered at p and a chart centered at f(p) in which f has the normal form (x^1, ..., x^m) -> (x^1, ..., x^k, 0, ..., 0), where m = dim M and the target sits in R^n. The proof is essentially the inverse function theorem applied twice: first straighten out the k independent input directions, then the k independent output directions. The hypothesis that the rank is constant — not just equal to k at the single point p — is essential; rank can only jump up nearby (it is lower semicontinuous), so demanding it stay put is a real condition.
Two famous special cases fall straight out. If rank equals dim M everywhere (an immersion), f locally looks like the inclusion (x) -> (x, 0), so locally it embeds M into N. If rank equals dim N everywhere (a submersion), f locally looks like the projection (x, y) -> (x), so its fibers are nicely stacked submanifolds and level sets are manifolds. The honest caveat: the theorem is purely local — it does not say f is globally an embedding or a fibration, only that around each point it is modeled on the standard map. Global injectivity, properness, and the like are separate hypotheses, which is exactly why an immersion need not be an embedding.
The map f: R^2 -> R^3, f(s, t) = (s, t, s + t), has differential of constant rank 2 everywhere. The theorem promises coordinates making it the standard inclusion (s, t) -> (s, t, 0); indeed the shear u = z - x - y already does it, recasting the image as the plane u = 0.
Constant rank 2 lets us straighten the map into the standard inclusion.
The hypothesis is constant rank on a neighborhood, not rank k at one point. Rank is lower semicontinuous, so it can only increase nearby; an isolated rank value gives you nothing.