Immersed submanifold - Imported Knowlpedia Content

Definition

Let $M$ be a smooth manifold. An immersed submanifold of $M$ is a smooth manifold $S$ together with an injective smooth immersion $\iota:S\to M$ . Thus each differential $d\iota_s:T_sS\to T_{\iota(s)}M$ is injective, but the topology and smooth structure on $S$ are part of the data and need not be induced from $M$ . When $\iota$ is understood, one often identifies $S$ set-theoretically with $\iota(S)$ , while retaining its given manifold topology. Its codimension is $\dim M-\dim S$ .

Local form

The constant-rank theorem implies that near each $s\in S$ , there are coordinates on $S$ and $M$ in which

\iota(x^1,\ldots,x^k)=(x^1,\ldots,x^k,0,\ldots,0).

Consequently every immersion is locally an embedding near each source point. This statement is source-local: distinct pieces of the source may accumulate in or pass through the same ambient region.

Comparison with embedded submanifolds

An injective immersion is an embedding exactly when it is a homeomorphism onto its image with the subspace topology. Hence every embedded submanifold is immersed, but not conversely. Proper injective immersions are embeddings, a useful sufficient condition recorded in Lee, chapters on immersions and submanifolds.

Examples and conventions

For irrational $\alpha$ , the map

t\longmapsto (e^{it},e^{i\alpha t})

injectively immerses $\mathbb{R}$ as a dense subset of the two-torus. It is not an embedding because its source topology is not the subspace topology of the dense image.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. DOI record. Relevant: Chapters 4–5.
M. W. Hirsch, Differential Topology, Springer, 1976. DOI record. Relevant: Chapter 1.