查看“︁浸入”︁的源代码

[[File:Klein bottle.svg|thumb|[[克萊因瓶]]浸入到3-空間中。]]
[[數學]]上，'''浸入'''是[[微分流形]]之間的[[可微映射]]，其[[前推 (微分)|導數]]處處是[[單射]]。確切而言，''f'' : ''M'' → ''N''是浸入，若在''M''中每一點''p''，

:<math>D_pf : T_p M \to T_{f(p)}N\,</math>

都是[[单射]]。（''T<sub>p</sub>X''表示''X''在點''p''處的[[切空間]]。另一個等價說法是''f''是浸入，若''f''的[[秩 (微分拓撲)|秩]]是常數，且等於''M''的維數：

:<math>\operatorname{rank}\,D_p f = \dim M.</math>

以上只要求''f''的導數為單射，但映射''f''未必是單射。

一個與浸入相關的概念是[[嵌入 (數學)#微分拓撲|嵌入]]。光滑嵌入是一個單射浸入''f'' : ''M'' → ''N''而同時為拓撲嵌入，使得''M''與其在''N''中的像[[微分同胚]]。浸入正是局部嵌入，即對''M''中每一點''x''都有一個''x''的[[鄰域]]''U'' ⊂ ''M''，使得''f'' : ''U'' → ''N''是嵌入。相反地，局部嵌入都是浸入。

[[File:Immersedsubmanifold nonselfintersection.jpg|thumb|一個單射[[子流形#浸入子流形|浸入子流形]]而不是嵌入。]]
若''M''是[[緊緻]]的，則單射浸入是一個嵌入；若''M''不是[[緊緻]]，則未必成立。這兩者的關係就如同連續雙射之於[[同胚]]。

==參考==
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==外部連結==
*[http://www.map.mpim-bonn.mpg.de/Immersion Immersion] {{Wayback|url=http://www.map.mpim-bonn.mpg.de/Immersion |date=20200809002453 }} at the Manifold Atlas
*[http://www.encyclopediaofmath.org/index.php/Immersion_of_a_manifold Immersion of a manifold] {{Wayback|url=http://www.encyclopediaofmath.org/index.php/Immersion_of_a_manifold |date=20150331073015 }} at the Encyclopedia of Mathematics

[[Category:微分拓扑学]]
[[Category:光滑函数]]