Is local vector bundle chart a homeomorphism, diffeomorphism, or bijection

differential-geometrydifferential-topologyvector-bundles

I've seen a few definitions, and each demands a local vector bundle chart map $\varphi:\pi^{-1}(U) \to U \times F$ to be something different: homeomorphism, diffeomorphism, or bijection.

E.g.:

  1. Wiki, the topology of fiber bundles Lecture notes Ralph L. Cohen – homeomorphism;
  2. Notes on Vector Bundles Aleksey Zinger – diffeomorphism;
  3. Foundations of mechanics 2nd edition R. Abraham and E. Marsden – bijection, but put additional constraint onto transition maps to be a diffeomorphisms.

I believe, there is a meaningful answer for such illusional ambiguity as it is for manifolds: see Verifying that a space is a Manifold via compatible charts

Best Answer

The concept of "vector bundle" has a topological variant and a smooth variant. It always depends on the context whether we work with topological or smooth vector bundles.

  1. Topological vector bundles.

Here we have a continuous map $p : E \to B$ between topological spaces $E, B$ and require the existence of local vector bundle charts $\varphi:\pi^{-1}(U) \to U \times F$ which are homeomorphisms. "Homeomorphism" is the maximum we can expect because there are no structures on $E, B$ beyond their topologies.

  1. Smooth vector bundles.

Here we have a smooth map $p : E \to B$ between smooth manifolds $E, B$ and require the existence of local vector bundle charts $\varphi:\pi^{-1}(U) \to U \times F$ which are diffeomorphisms. The smooth structures of $E, B$ allow the stronger "diffeomorphism" requirement. Clearly each smooth vector bundle is a topological vector bundle. But there are topological vector bundles $p : E \to B$ with smooth manifolds $E, B$ which are no smooth vector bundles.

What is difference between the above concept of smooth vectoer bundle and Abraham and Marsden's approaach in "Foundations of Mechanics"?

They do not start with a smooth map $p : E \to B$, but with a set $S$ which has a vector bundle atlas $\mathscr B$ consisting of local bundle charts on $S$. Since $S$ does not yet have a topology, such bundle charts can only be required to be bijections. However, in a vector bundle atlas the transition maps are required to be diffeomorphisms. This can be done because the transition functions are maps between open subsets of Euclidean spaces.

This allows to give $S$ a unique topology making all local bundle charts in $\mathscr B$ homeomorphisms. Doing so makes $\mathscr B$ a smooth atlas on $S$. Thus $S$ receives the structure of a smooth manifold denoted by $E$. This procedure is discussed in Verifying that a space is a Manifold via compatible charts.

There is still no map $p : E \to B$ to a smooth manifold $B$, but the authors introduce the zero-section $E_0 \subset E$ which is a smooth submanifold of $E$ and a map $\pi : E \to E_0$ which is smooth. It is then easy to see that $\pi : E \to E_0$ is a smooth vector bundle in the sense of the above definition.

We can therefore say that Abraham and Marsden have chosen a constructive approach: They build smooth vector bundles by smoothly patching together local vector bundles of the form $U \times F$. Their approach results in the same concept of a vector bundle as in the above definition. For practical purposes this constructive approach is in some sense superior. Many standard smooth vector bundles (as the tangent bundle $TM$ of a smooth manifold $M$) arise by first defining its underlying set which is given the structure of a smooth vector bundle using a vector bundle atlas in the above sense.

Related Question