Difference between revisions of "Parallel transport"
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==Formal definition of a topological bundle== | ==Formal definition of a topological bundle== | ||
| − | Let $\pi:E\to B$ be a continuous map between topological spaces, called the ''total space'' and the ''base'', and $F$ yet another topological space called ''fiber'', such that the preimage $F_b=\ | + | Let $\pi:E\to B$ be a continuous map between topological spaces, called the ''total space'' and the ''base'', and $F$ yet another topological space called ''fiber'', such that the preimage $F_b=\pi^{-1}(b)\subset E$ of every point of the base is [[homeomorphism|homeomorphic]] to $X$. The latter condition means that $E$ is the disjoint union of "fibers", $E=\bigsqcup_{b\in B} F_b$ homeomorphic to each other. |
The map $\pi$ is called [[fibration]] of $E$ over $B$, if the above representation is ''locally trivial'': any point of the base admits an open neighborhood $U$ such that the restriction of $\pi$ on the preimage $\pi^{-1}(U)$ is [[topologically equivalence|topologically equivalent]] to the Cartesian projection $\pi_2$ of the product $F\times U$ on the second component: $\pi_2(v,b)=b$. Formally this means that there exists a homeomorhism $H_U=H:\pi^{-1}(U)\to F\times U$ such that $\pi=\pi_2\circ H$. | The map $\pi$ is called [[fibration]] of $E$ over $B$, if the above representation is ''locally trivial'': any point of the base admits an open neighborhood $U$ such that the restriction of $\pi$ on the preimage $\pi^{-1}(U)$ is [[topologically equivalence|topologically equivalent]] to the Cartesian projection $\pi_2$ of the product $F\times U$ on the second component: $\pi_2(v,b)=b$. Formally this means that there exists a homeomorhism $H_U=H:\pi^{-1}(U)\to F\times U$ such that $\pi=\pi_2\circ H$. | ||
Revision as of 06:56, 7 May 2012
A very flexible construction aimed to represent a family of similar objects (fibres or fibers, depending on the preferred spelling) which are parametrized by the index set which itself has an additional structure (topological space, smooth manifold etc.).
The most known examples are the tangent and cotangent bundles of a smooth manifold.
Formal definition of a topological bundle
Let $\pi:E\to B$ be a continuous map between topological spaces, called the total space and the base, and $F$ yet another topological space called fiber, such that the preimage $F_b=\pi^{-1}(b)\subset E$ of every point of the base is homeomorphic to $X$. The latter condition means that $E$ is the disjoint union of "fibers", $E=\bigsqcup_{b\in B} F_b$ homeomorphic to each other.
The map $\pi$ is called fibration of $E$ over $B$, if the above representation is locally trivial: any point of the base admits an open neighborhood $U$ such that the restriction of $\pi$ on the preimage $\pi^{-1}(U)$ is topologically equivalent to the Cartesian projection $\pi_2$ of the product $F\times U$ on the second component: $\pi_2(v,b)=b$. Formally this means that there exists a homeomorhism $H_U=H:\pi^{-1}(U)\to F\times U$ such that $\pi=\pi_2\circ H$.
Parallel transport. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallel_transport&oldid=26147