Immersions

Immersions

An immersion is a smooth map f from one smooth manifold M to another smooth manifold N such that its differential is injective at every point. If dim M = m and dim N = n, this means that m ≤ n and the linear map df_p: T_pM → T_{f(p)}N has rank m for all p ∈ M. Locally, immersions resemble the standard inclusion of Euclidean space: around each p there are charts in which f appears as (x1, ..., xm) ↦ (x1, ..., xm, 0, ..., 0). Therefore the image f(M) is a submanifold of N of dimension m, but f need not be injective, so immersions can have self-intersections.

Examples and intuition: The canonical inclusion R^k → R^n, x ↦ (x, 0) is an immersion. Regular parametrizations of curves in the plane are immersions of 1-manifolds into R^2. A classic illustration is a figure-eight curve in the plane, which is an immersion but not an embedding because it intersects itself.

Relation to other notions: An immersion is not necessarily an embedding; an embedding is an immersion that is also a homeomorphism onto its image (with the subspace topology). Immersions are related to submersions in that a submersion has a surjective differential at every point, opposite in rank behavior. Notable results in differential topology include Whitney’s immersion theorem, which states that every smooth m-manifold can be immersed into R^{2m−1}, and Whitney’s embedding theorem, which guarantees an embedding into R^{2m}.