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Fredrik

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I was asked what a bundle structure is in a PM. I'm posting my reply here, to give others a chance to add comments.

I didn't get to see the exact sentence in which the term "bundle structure" was used. I would guess that it was used informally to say that some smooth manifold is the total space of a fiber bundle, so the term that really needs to be explained is "fiber bundle".

The basic idea is this: Consider a function ##\pi:E\to B## that's surjective onto ##B##. The sets ##\pi^{-1}(b)## with ##b\in B## are mutually disjoint, and their union is ##E##. It's convenient to call each ##\pi^{-1}(b)## the "fiber over ##b##" and to think of ##E## as a "bundle" of fibers. ##E## is also called the "total space". ##\pi## is called "the projection". If there's a set ##F## such that each ##\pi^{-1}(b)## can be mapped bijectively onto ##F##, then ##F## is called "the fiber".

This concept is however pretty useless when ##E## and ##B## are just sets. The term "fiber bundle" is usually only defined when ##E,B,F## are structures of the same type, e.g. when they're all topological spaces or all smooth manifolds. Then we make additional requirements on the relationship between ##F## and the fibers ##\pi^{-1}(b)## that's appropriate for the type of structure we're dealing with. If we're dealing with topological spaces, then we require that ##F## is homeomorphic to each of the ##\pi^{-1}(b)##. If we're dealing with smooth manifolds, then we require that ##F## is diffeomorphic to each of the ##\pi^{-1}(b)##.

The 4-tuple ##(E,B,F,\pi)## is then called a "fiber bundle". A nice example is when E is a cylinder, B is a circle and F is a line segment. Another nice example is when E is a Möbius strip, B is a circle and F is a line segment.

Note that if ##\{U_i|i\in I\}## is a collection of open subsets of ##B## that covers ##B##, then ##\{\pi^{-1}(U_i)|i\in I\}## is a collection of subsets of ##E## that covers ##E##. The total space ##E## can be thought of as consisting of the overlapping pieces ##\pi^{-1}(U_i)## that are glued together. In the case when B is a circle and F is a line segment, we can pick two open sets ##U,V## such that ##B=U\cup V##. The sets ##\pi^{-1}(U)## and ##\pi^{-1}(V)## will be like two rectangular strips of paper that are glued together to form the space ##E##, and as we all know two rectangular strips of paper can be glued together to form either a circle or a Möbius strip.

The most useful fiber bundle is the tangent bundle ##TM## of a smooth manifold ##M##. There are a few slightly different ways to define it. One is to simply take ##TM## to be the union of all the tangent spaces, i.e. ##TM=\bigcup_{p\in M} T_pM##. The base space is the smooth manifold ##M##. The projection is the map ##\pi:TM\to M## defined by saying that for each ##v\in TM##, ##\pi(v)## is the unique ##p\in M## such that ##v\in T_pM##.

A

I didn't get to see the exact sentence in which the term "bundle structure" was used. I would guess that it was used informally to say that some smooth manifold is the total space of a fiber bundle, so the term that really needs to be explained is "fiber bundle".

The basic idea is this: Consider a function ##\pi:E\to B## that's surjective onto ##B##. The sets ##\pi^{-1}(b)## with ##b\in B## are mutually disjoint, and their union is ##E##. It's convenient to call each ##\pi^{-1}(b)## the "fiber over ##b##" and to think of ##E## as a "bundle" of fibers. ##E## is also called the "total space". ##\pi## is called "the projection". If there's a set ##F## such that each ##\pi^{-1}(b)## can be mapped bijectively onto ##F##, then ##F## is called "the fiber".

This concept is however pretty useless when ##E## and ##B## are just sets. The term "fiber bundle" is usually only defined when ##E,B,F## are structures of the same type, e.g. when they're all topological spaces or all smooth manifolds. Then we make additional requirements on the relationship between ##F## and the fibers ##\pi^{-1}(b)## that's appropriate for the type of structure we're dealing with. If we're dealing with topological spaces, then we require that ##F## is homeomorphic to each of the ##\pi^{-1}(b)##. If we're dealing with smooth manifolds, then we require that ##F## is diffeomorphic to each of the ##\pi^{-1}(b)##.

The 4-tuple ##(E,B,F,\pi)## is then called a "fiber bundle". A nice example is when E is a cylinder, B is a circle and F is a line segment. Another nice example is when E is a Möbius strip, B is a circle and F is a line segment.

Note that if ##\{U_i|i\in I\}## is a collection of open subsets of ##B## that covers ##B##, then ##\{\pi^{-1}(U_i)|i\in I\}## is a collection of subsets of ##E## that covers ##E##. The total space ##E## can be thought of as consisting of the overlapping pieces ##\pi^{-1}(U_i)## that are glued together. In the case when B is a circle and F is a line segment, we can pick two open sets ##U,V## such that ##B=U\cup V##. The sets ##\pi^{-1}(U)## and ##\pi^{-1}(V)## will be like two rectangular strips of paper that are glued together to form the space ##E##, and as we all know two rectangular strips of paper can be glued together to form either a circle or a Möbius strip.

The most useful fiber bundle is the tangent bundle ##TM## of a smooth manifold ##M##. There are a few slightly different ways to define it. One is to simply take ##TM## to be the union of all the tangent spaces, i.e. ##TM=\bigcup_{p\in M} T_pM##. The base space is the smooth manifold ##M##. The projection is the map ##\pi:TM\to M## defined by saying that for each ##v\in TM##, ##\pi(v)## is the unique ##p\in M## such that ##v\in T_pM##.

A

*section*of the tangent bundle ##TM## is a function ##X:U\to TM## such that ##U\subseteq B## and ##X(p)\in T_pM## for all ##p\in U##. A section of ##TM## is also called a "vector field". (Edit: I fixed a mistake in this paragraph after it was pointed out by Orodruin below).
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