Fiber of a map

Let $f:X\to Y$ be a map of sets and let $y\in Y$. The fiber of $f$ over $y$ (or preimage fiber) is the subset

If $f:M\to N$ is a smooth map between smooth manifolds , the fiber $f^{-1}(y)$ is still defined as a subset of $M$. Additional geometry arises when $y$ is a regular value in the sense of the differential of a smooth map : in that case, $f^{-1}(y)$ is an embedded submanifold of $M$ of codimension $\dim N$.

Fibers are especially important when $f$ is a bundle projection; for instance, in a principal G-bundle $\pi:P\to B$, each fiber $\pi^{-1}(b)$ is (noncanonically) diffeomorphic to the structure group $G$.

Examples

1. Projection of a product. For $\pi:M\times F\to M$, the fiber over $m\in M$ is $\pi^{-1}(m)=\{m\}\times F$, canonically identified with $F$.
2. Radius-squared map. For $f:\mathbb{R}^2\to\mathbb{R}$, $f(x,y)=x^2+y^2$, the fiber over $r$ is empty if $r<0$, is a single point if $r=0$, and is a circle of radius $\sqrt r$ if $r>0$.
3. Hopf fibration (geometric fiber). The Hopf map $S^3\to S^2$ has fibers diffeomorphic to $S^1$; it is a principal $S^1$-bundle, so every fiber is an orbit of the $S^1$-action.