CONNECTIONS AND COVARIANT DERIVATIVES ON VECTOR BUNDLES

VICTOR SANTOS

1. Fibre bundles

Let B, E, F be smooth manifolds, together with a smooth surjective projection p : E −→ B. Wesay that p has the local product property in u ∈ B with respect to F if there is an open set U ⊂ B

containing u and a smooth diffeomorphism ψ : p−1(U) −→ U × F such that the diagram

p−1(U) U × F

U

ψ

p

π1

commutes. The map π1 : U × F −→ U is the projection onto the first factor: π1(u, f) = u, for anyu ∈ U and f ∈ F . The set U is called a trivializing chart and ψ a trivializing diffeomorphism for U .The pair (U, ψ) is called a local trivializing representation.

(E, p,B, F ) is called a fibre bundle over B if p has the local product property with respect to F forany point u ∈ B. E is the total space, B the base space and F is the typical fibre of the bundle. Theset Eu = p−1(u) is called the fibre over u and it is a closed smooth submanifold of E, diffeomorphicto F for any u ∈ B.

2. Vector bundles

A smooth fibre bundle (E, p, B,E) is called a vector bundle is E and Eu = p−1(u) are real vectorspaces for all u ∈ B, and if there is a covering collection of trivializing representations (Uα, ψα) suchthat each trivializing diffeomorphism ψα : p−1(Uα) −→ Uα × E is fibrewise linear; that is, the mapψα,u : Eu −→ E given by ψα,u = π2 ◦ ψα|Eu

is a linear isomorphism. A smooth map s : B −→ E iscalled a smooth section of E iff p ◦ s = idB. We denote the space of all smooth sections by Γ(E).

3. Connections

In a smooth vector bundle (E, p,M,E), the map p induces the tangent map Tp(u) : TuE −→Tp(u)M . The vertical bundle of E, denoted by VE, is the kernel of Tp:

VE = kerTp.

A subdundle HE of the tangent bundle TE is called a horizontal bundle over E is the Whitney sumVE⊕HE is strongly isomorphic to TE:

TE = VE⊕HE;

then, if HuE denotes the fibre of the horizontal bundle over u ∈ E we have Tp(u) : HuE −→ Tp(u)M .A connection on the vector bundle is a smooth choice of vertical vectors, given by the map

Φ : E −→ VE,

such that Φ ◦ Φ = Φ and ImΦ = VE. With this definition the horizontal bundle is simply given byHE = kerΦ.

Let N be a smooth manifold and f : N −→ E. Then for any v ∈ TqN , T (Φ ◦ f)(q)(v) ∈ V(Φ◦f)(q)E,and since VuE is canonically isomorphic to Ep(u), T (Φ ◦ f)(q)(v) can be viewed as a vector in Ep(f(q)),which we denote by ∇vf :

∇vf = T (Φ ◦ f)(q)(v).1

2 VICTOR SANTOS

This yields the smooth map ∇f : TN −→ E; in particular for any X ∈ Γ(TM) and any s ∈ Γ(E)yield ∇Xs ∈ Γ(E), defined by

(∇Xs)(u) = ∇X(u)s ∈ Eu

for all u ∈M . The map

∇ :Γ(TM)× Γ(E) −→ Γ(E)

∇(X, s) 7→ ∇Xs

has the following properties:

(1)

∇X(s1 + s2)(u) = T (Φ ◦ (s1 + s2))(u)(X)

= T (Φ ◦ s1 + Φ ◦ s2)(u)(X)

= T (Φ ◦ s1)(u)(X) + T (Φ ◦ s2)(u)(X)

= ∇Xs1(u) +∇Xs2(u)

(2)

∇X(τ · s) = T (Φ ◦ (τ · s))(u)(X)

= T (τ · (Φ ◦ s))(u)(X) MISSING PROOF !!

(3)

∇X+Y s(u) = T (Φ ◦ s)(u)(X + Y )

= T (Φ ◦ s)(u)(X) + T (Φ ◦ s)(u)(Y )

= ∇Xs+∇Y s

(4)

∇τXs(u) = T (Φ ◦ s)(u)(τX)

= τT (Φ ◦ s)(u)(X)

= τ∇Xs

This map is called a covariant derivative. A covariant derivative defines a curvature operator

R : Γ(TM)× Γ(TM)× Γ(E) −→ Γ(E)

which assigns to each pair X, Y ∈ Γ(TM) and each s ∈ Γ(E) the section

R(X, Y )s := ∇X∇Y s−∇Y∇Xs−∇[X,Y ]s ∈ Γ(E)

4. Scalar field in Minkowski space

In order to construct the scalar field model, we start from the vector bundle (E, p,M, V ), whereE, V are inner product spaces and M is a Lorentzian manifold with metric η