immersion Immersion 2002-04-15 17:44:34 2004-12-11 01:10:17 Definition closed immersion % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 $df_x: T_x(X) \to T_y(Y)$ is the derivative of $f$ at $x$, and $T_z(Z)$ is the tangent space of manifold $Z$ at point $z$. If $df_x$ is injective, then $f$ is said to be an \emph{immersion at x}. If $f$ is an immersion at every point, it is called an \emph{immersion}. If the image of $f$ is also closed, then $f$ is called a \emph{closed immersion}. The notion of \PMlinkname{closed immersion}{ClosedImmersion} for schemes is the analog of this notion in algebraic geometry.