Vector Bundle - Smooth Vector Bundles

Smooth Vector Bundles

A vector bundle (E,p,M) is smooth, if E and M are smooth manifolds, p : E → M is a smooth map, and the local trivializations are diffeomorphisms. Depending on the required degree of smoothness, there are different corresponding notions of Cp bundles, infinitely differentiable C∞-bundles and real analytic Cω-bundles. In this section we will concentrate on C∞-bundles. The most important example of a C∞-vector bundle is the tangent bundle (TM,πTM,M) of a C∞-manifold M.

The C∞-vector bundles (E,p,M) have a very important property not shared by more general C∞-fibre bundles. Namely, the tangent space Tv(Ex) at any v∈Ex can be naturally identified with the fibre Ex itself. This identification is obtained through the vertical lift vlv:Ex→Tv(Ex), defined as

\operatorname{vl}_vw := \frac{d}{dt}\Big|_{t=0}f(v+tw), \quad f\in C^\infty(E_x).

The vertical lift can also be seen as a natural C∞-vector bundle isomorphism p*E→VE, where (p*E,p*p,E) is the pull-back bundle of (E,p,M) over E through p:E→M, and VE:=Ker(p*)⊂TE is the vertical tangent bundle, a natural vector subbundle of the tangent bundle (TE,πTE,E) of the total space E.

The slit vector bundle E/0, obtained from (E,p,M) by removing the zero section 0⊂E, carries a natural vector field Vv := vlvv, known as the canonical vector field. More formally, V is a smooth section of (TE,πTE,E), and it can also be defined as the infinitesimal generator of the Lie-group action

\Phi_V:\mathbb R \times (E\setminus 0) \to (E\setminus 0) \quad ; \quad (t,v)\mapsto \Phi_V^t(v) := e^tv.

For any smooth vector bundle (E,p,M) the total space TE of its tangent bundle (TE,πTE,E) has a natural secondary vector bundle structure (TE,p*,TM), where p* is the push-forward of the canonical projection p:E→M. The vector bundle operations in this secondary vector bundle structure are the push-forwards +*:T(E×E)→TE and λ*:TE→TE of the original addition +:E×E→E and scalar multiplication λ:E→E.

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