In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if
D p f : T p M → T f ( p ) N {\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,}
is an injective function at every point p of M (where TpX denotes the tangent space of a manifold X at a point p in X and Dp f is the derivative (pushforward) of the map f at point p). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M:
rank D p f = dim M . {\displaystyle \operatorname {rank} \,D_{p}f=\dim M.}
The function f itself need not be injective, only its derivative must be.
Vs. embedding A related concept is that of an embedding. A smooth embedding is an injective immersion f : M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, for any point x ∈ M there is a neighbourhood, U ⊆ M, of x such that f : U → N is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes taken to be the definition of an immersion.
If M is compact, an injective immersion is an embedding, but if M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.
Regular homotopy A regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function H : M × [0,1] → N such that for all t in [0, 1] the function Ht : M → N defined by Ht(x) = H(x, t) for all x ∈ M is an immersion, with H0 = f, H1 = g. A regular homotopy is thus a homotopy through immersions.
Classification Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for 2m < n + 1 every map f : M m → N n of an m-dimensional manifold to an n-dimensional manifold is homotopic to an immersion, and in fact to an embedding for 2m < n; these are the Whitney immersion theorem and Whitney embedding theorem. Stephen Smale expressed the regular homotopy classes of immersions f : M m → R n {\displaystyle f:M^{m}\to \mathbb {R} ^{n}} as the homotopy groups of a certain Stiefel manifold. The sphere eversion was a particularly striking consequence. Morris Hirsch generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any m-dimensional manifold M m in any n-dimensional manifold N n. The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov.
Existence
… excerpt ends here. Continue reading the full article.
