immersions, submersions, and embeddings

These three words classify smooth maps by what their differential does to dimensions, and they are the vocabulary for talking about how one manifold sits inside another. An immersion injects directions (no infinitesimal collapsing), a submersion surjects directions (it spreads out onto the target), and an embedding is an immersion that also lays the manifold down cleanly with no self-crossings or wild accumulation.

Let f: M -> N be smooth. It is an immersion if df_p is injective at every p (so rank equals dim M, forcing dim M at most dim N); it is a submersion if df_p is surjective at every p (rank equals dim N, forcing dim M at least dim N). An embedding is an immersion that is also a homeomorphism onto its image with the subspace topology — equivalently an injective immersion that is proper, or whose image carries the correct topology. The constant-rank theorem gives the local pictures: an immersion locally looks like the inclusion R^k -> R^n, a submersion locally like the projection R^m -> R^n.

The gap between an immersion and an embedding is the whole story of subtlety here. The figure-eight curve is the image of an injective immersion of an open interval, but it is not an embedded submanifold: the immersion is not a homeomorphism onto its image because the two ends of the interval limit onto the crossing point, so the subspace topology disagrees with the interval topology. The dense line of irrational slope on the torus is an injective immersion that is not even injective-with-good-topology — its image is dense, not a submanifold at all. So injectivity plus immersion is not enough for embedding; you genuinely need the topological condition, usually supplied in practice by properness or compactness of the domain.