浸入

數學上，浸入是微分流形之間的可微映射，其導數處處是單射。確切而言，f : M → N是浸入，若在M中每一點p，

D_{p}f:T_{p}M\to T_{f(p)}N\,

都是單射。（T_pX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的秩是常數，且等於M的維數：

\operatorname {rank} \,D_{p}f=\dim M.

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。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。

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

Adachi, Masahisa, Embeddings and immersions, 1993 [2013-09-29], ISBN 978-0-8218-4612-4, （原始內容存檔於2013-11-26）, translation Kiki Hudson
Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M., Singularities of Differentiable Maps: Volume 1, Birkhäuser, 1985, ISBN 0-8176-3187-9
Bruce, J. W.; Giblin, P. J., Curves and Singularities, Cambridge University Press, 1984, ISBN 0-521-42999-4
Carter, J. Scott; Saito, Masahico, Surfaces in 3-Space That Do Not Lift to Embeddings in 4-Space, 1995 [2013-09-29], （原始內容存檔於2016-03-04）published in conference proceedings Knot theory, Banach center publications, 42 Warzawa (1998), 29–47.
Carter, J. Scott; Saito, Masahico, Knotted Surfaces and Their Diagrams, Mathematical Surveys and Monographs 55: 258, 1998, ISBN 978-0-8218-0593-0
Carter, J. Scott; Kamada, Seiichi; Saito, Masahico, Surfaces in 4-space, 2004
Gromov, M., Partial differential relations, Springer, 1986, ISBN 3-540-12177-3
Hirsch M. Immersions of manifolds. Trans. A.M.S. 93 1959 242—276.
Koschorke, Ulrich, Multiple points of Immersions and the Kahn-Priddy Theorem, Math Z., 1979, (169): 223–236
Smale, S. A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90 1958 281–290.
Smale, S. The classification of immersions of spheres in Euclidean spaces. Ann. of Math. (2) 69 1959 327—344.
Spring, D., The Golden Age of Immersion Theory in Topology: 1959-1973 (PDF), Bulletin of the American Mathematical Society, 2005, (42): 163–180 [2013-09-29], （原始內容存檔 (PDF)於2008-07-25）
Wall, C. T. C.: Surgery on compact manifolds. 2nd ed., Mathematical Surveys and Monographs 69, A.M.S.

外部連結

Immersion （頁面存檔備份，存於互聯網檔案館） at the Manifold Atlas
Immersion of a manifold （頁面存檔備份，存於互聯網檔案館） at the Encyclopedia of Mathematics