v bundle
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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