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,π_TM,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 T_v(E_x) at any v∈E_x can be naturally identified with the fibre E_x itself. This identification is obtained through the vertical lift vl_v:E_x→T_v(E_x), 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,π_TE,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 V_v := vl_vv, known as the canonical vector field. More formally, V is a smooth section of (TE,π_TE,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,π_TE,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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Famous quotes containing the words smooth and/or bundles:

“Nations are possessed with an insane ambition to perpetuate the memory of themselves by the amount of hammered stone they leave. What if equal pains were taken to smooth and polish their manners?”
—Henry David Thoreau (1817–1862)

“He bundles every forkful in its place,
And tags and numbers it for future reference,
So he can find and easily dislodge it
In the unloading. Silas does that well.
He takes it out in bunches like birds’ nests.”
—Robert Frost (1874–1963)