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