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 $df_{x}:T_{x}(X)\\to T_{y}(Y)$ 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 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.

单词	Immersion
释义	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 $df_{x}:T_{x}(X)\\to T_{y}(Y)$ 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 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.
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