Immersion (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Immersion_(mathematics)


00:00:34 1 Regular homotopy
00:00:51 2 Classification
00:01:09 2.1 Existence
00:01:43 2.2 Codimension 0
00:02:18 3 Multiple points
00:02:53 4 Examples and properties
00:03:27 4.1 Immersed plane curves
00:04:02 4.2 Immersed surfaces in 3-space
00:04:37 5 Generalizations
00:05:11 6 See also
00:05:46 7 Notes
00:06:21 8 References



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SUMMARY
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For a closed immersion in algebraic geometry, see closed immersion.In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is an immersion if

  
    
      
        
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    {\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). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M:

  
    
      
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    {\displaystyle \operatorname {rank} \,D_{p}f=\dim M.}
  The function f itself need not be injective, only its derivative.
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 – i.e., 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.