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.

Read more about this topic:  Vector Bundle

Famous quotes containing the words smooth and/or bundles:

    “he changed and ran
    Through many shapes; I lunged at the smooth throat
    Of a great eel; it changed, and I but smote
    A fir-tree roaring in its leafless top;
    And thereupon I drew the livid chop
    Of a drowned dripping body to my breast....”
    —William Butler Yeats (1865–1939)

    “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)