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，
都是单射。（TpX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：
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。
參考 编辑
- Adachi, Masahisa, Embeddings and immersions, 1993 [2013-09-29], ISBN 978-0-8218-4612-4, （原始内容存档于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], （原始内容存档于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], （原始内容存档 (PDF)于2008-07-25）
- Wall, C. T. C.: Surgery on compact manifolds. 2nd ed., Mathematical Surveys and Monographs 69, A.M.S.