# immersion

Let $X$ and $Y$ be manifolds^{}, and let $f$ be a mapping $f:X\to Y$. Choose $x\in X$, and let $y=f(x)$. Recall that $d{f}_{x}:{T}_{x}(X)\to {T}_{y}(Y)$ is the derivative of $f$ at $x$, and ${T}_{z}(Z)$ is the tangent space of manifold $Z$ at point $z$.

If $d{f}_{x}$ is injective, then $f$ is said to be an *immersion at x*. If $f$ is an immersion at every point, it is called an *immersion*.

If the image of $f$ is also closed, then $f$ is called a *closed immersion*.

The notion of closed immersion (http://planetmath.org/ClosedImmersion) for schemes is the analog of this notion in algebraic geometry^{}.

Title | immersion |
---|---|

Canonical name | Immersion |

Date of creation | 2013-03-22 12:35:04 |

Last modified on | 2013-03-22 12:35:04 |

Owner | bshanks (153) |

Last modified by | bshanks (153) |

Numerical id | 8 |

Author | bshanks (153) |

Entry type | Definition |

Classification | msc 57R42 |

Related topic | Submersion^{} |

Defines | closed immersion |