# 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.

Title immersion Immersion 2013-03-22 12:35:04 2013-03-22 12:35:04 bshanks (153) bshanks (153) 8 bshanks (153) Definition msc 57R42 Submersion closed immersion