Fig. 2.17 Two local charts of the base manifold M yield two local trivializations of the tangent bundle T M

2.5.1 Sections of a Bundle

It is now the appropriate time to associate a precise definition to the notion of bundle section that we have implicitly advocated in (2.5.2).

π

Definition 2.5.1 Consider a generic fibre-bundle E −→ M with generic fibre F . We name section of the bundle a rule s that to each point p M of the base manifold associates a point s(p) Fp in the fibre above p, namely a map

The above definition is illustrated in Fig. 2.18 which also clarifies the intuitive idea standing behind the chosen name for such a concept.

It is clear that sections of the bundle can be chosen to be continuous, differentiable, smooth or, in the case of complex manifolds, even holomorphic, depending on the properties of the map s in each local trivialization of the bundle. Indeed given a local trivialization and given open charts for both the base manifold M and for the fibre F , the local description of the section reduces to a map:

where m and n are the dimensions of the base manifold and of the fibre respectively. We are specifically interested in smooth sections, namely in section that are in-

π

finitely differentiable. Given a bundle E −→ M , the set of all such sections is denoted by:

Of particular relevance are the smooth sections of vector bundles. In this case to each point of the base manifold p we associate a vector v(p) in the vector space above the point p. In particular we can consider sections of the tangent bundle T M associated with a smooth manifold M . Such sections correspond to the notion of vector fields.

Fig. 2.18 A section of a fibre bundle

Definition 2.5.2 Given a smooth manifold M , we name vector field on M a smooth section t Γ (T M , M ) of the tangent bundle. The local expression of such vector field in any open chart (U, φ) is

2.5.1.1 Example: Holomorphic Vector Fields on S2

As we have seen above, the 2-sphere S2 is a complex manifold of complex dimension one covered by an atlas composed by two charts, that of the North Pole and that of the South Pole (see Fig. 2.19) and the transition function between the local complex coordinate in the two patches is the following one:

Correspondingly, in the two patches, the local description of a holomorphic vector field t is given by:

d t = vN (zN ) dzN

(2.5.13)

d t = vS (zS ) dzS

where the two functions vN (zN ) and vS (zS ) are supposed to be holomorphic functions of their argument, namely to admit a Taylor power series expansion:

∞

vN (zN ) = ck zNk

k=0

(2.5.14)

∞

vS (zS ) = vS (zS ) dk zSk

k=0

However, from the transition function (2.5.12) we obtain the relations: