3.4 Connections on a Vector Bundle

3.4 Connections on a Vector Bundle

π

Let us now consider a generic vector bundle E = M of rank r. Its standard fi- bre is an r-dimensional vector space V and the transition functions from one local trivialization to another one are maps:

#

ψαβ : Uα Uβ → Hom(V, V) GL(r, R) (3.4.1)

In other words, without further restrictions the structural group of a generic vector bundle is just GL(r, R).

The notion of a connection on generic vector bundles is formulated according to the following:

π π

Definition 3.4.1 Let E = M be a vector bundle, T M = M the tangent bundle to the base manifold M (dim M = m) and let Γ (E, M ) be the space of sections of E: a connection on E is a rule that, to each vector field X Γ (T M , M ) associates a map:

satisfying the following defining properties:

(a)X(a1s1 + b1s2) = a1 Xs1 + a2 Xs2

(b)a1X1+a2X2 s = a1 X1 s + a2 X2 s

(c)X(f · s) = X(f ) · s + f · Xs

(d)f ·Xs = f · Xs

where a1,2 R are real numbers, s1,2 Γ (E, M ) are section of the vector bundle and f C∞(M ) is a smooth function.

The abstract description of a connection provided by Definition 3.4.1 becomes more explicit if we consider bases of sections for the two involved vector bundles.

π

To this effect let {si (p)} be a basis of sections of the vector bundle E = M (i = 1, 2, . . . , r = rank E) and let {eμ(p)}, (μ = 1, 2, . . . , m = dim M ) be a basis

π

of sections for the tangent bundle to the base manifold T M = M . Then we can write:

where the local functions Θμi j (p), ( p M ) are named the coefficients of the connection. They are necessary and sufficient to specify the connection on an arbitrary section of the vector bundle. To this effect let s Γ (E, M ). By definition of basis of sections we can write:

and correspondingly we obtain:

μs = μci · si

(3.4.5)

μci ≡ ∂μci + Θμj i cj

having defined ∂μci ≡ eμ(ci ). Confronting this with the discussions at the end of Sect. 3.3.2.1, we see that in the language of physicists μci can be identified as the covariant derivatives of the vector components ci . Let us now observe that for any

π

vector bundle E = M if:

r ≡ rank(E)

is the rank, namely the dimension of the standard fibre and:

m ≡ dim M

is the dimension of the base manifold, the connection coefficients can be viewed as a set of m matrices, each of them r × r:

Hence more abstractly we can say that the connection on a vector bundle associates to each vector field defined on the base manifold a matrix of dimension equal to the rank of the bundle:

X Γ (T M , M ) :

: X → ΘX = r × r-matrix depending on the base-point p M (3.4.7)

The relevant question is the following: can we relate the matrix ΘX to a one-form and make a bridge between the above definition of a connection and that provided by the Ehresmann approach? The answer is yes and it is encoded in the following equivalent:

bundle to the base manifold M (dim M = m) and let Γ (E, M ) be the space of sections of E: a connection on E is a map:

Γ(E, M ) such that the following defining properties are satisfied:

(a)(a1s1 + b1s2) = a1 s1 + a2 s2

(b)(f · s) = df · s + f · s

where a1,2 R are real numbers, s1,2 Γ (E, M ) are section of the vector bundle and f C∞(M ) is a smooth function.

The relation between the two definitions of a connection on a vector bundle is now easily obtained by stating that for each section s Γ (E, M ) and for each vector field X Γ (T M , M ) we have:

The reader can easily convince himself that using (3.4.9) and relying on the properties of s established by Definition 3.4.2 those of X s required by Definition 3.4.1 are all satisfied.

Consider now, just as before, a basis of sections of the vector bundle {si (p)}. According to its second Definition 3.4.2 a connection singles out an r × r matrixvalued one-form Θ Γ (T M , M ) through the equation:

Clearly the connection coefficients introduced in (3.4.7) are nothing else but the values of Θ on each vector field X:

A natural question which arises at this point is the following. In view of our comments at the end of Sect. 3.3.2.1 could we identify Θ with the matrix representation of A , the principal connection on the corresponding principal bundle? In other words could we write:

For a single generic vector bundle this question seems tautological. Indeed the structural group is simply GL(r, R) and being the corresponding Lie algebra gl(r, R) made by the space of generic matrices it looks like obvious that Θ could be viewed as Lie algebra valued. The real question however is another. The identification (3.4.12) is legitimate if and only if Θ transforms as the matrix representation of A . This is precisely the case. Let us demonstrate it. Consider the intersection of two-

"

local trivializations. On Uα Uβ the relation between two bases of sections is given by:

where t(αβ) is the transition function seen as an abstract group element. Equation (3.4.13) implies:

which is consistent with the gauge transformation rule (3.3.89) if we identify Θ as in (3.4.12).

The outcome of the above discussion is that once we have defined a connection one-form A on a principle bundle P (M , G) a connection is induced on any

π

associated vector bundle E = M . It is simply defined as follows:

where D() denotes the linear representation of both G and G that define the associated bundle.