The tri-index symbols Γμνρ encode, patch by patch, the considered affine connection. According to Definition 3.7.1 these connection coefficients are equivalently defined by setting

3.7.2 Curvature and Torsion of an Affine Connection

To every connection one-form A on a principal bundle P (M , G), we can associate a curvature 2-form:

which is G Lie algebra valued and whose fundamental properties and profound physical meaning will be analyzed in Chap. 5 of this volume. Here we just note that,

π

evaluated on any associated vector bundle E = M , the connection A becomes a matrix and the same is true of the curvature F . In that case the first line of (3.7.6) is to be understood in the sense both of matrix multiplication and of wedge product, namely the element (i, j ) of A A is calculated as Ai k Ak j , with summation over the dummy index k.

We can apply the general formula (3.7.6) to the case of an affine connection. In that case the curvature 2-form is traditionally denoted with the letter R in honor of

which, using the basis {eμ} for the tangent bundle and its dual {ων } for the cotangent bundle, becomes:

Comparing (3.7.9) with the Riemann-Christoffel symbols of (3.2.10), we see that the latter could be identified with the components of the curvature two-form of an affine connection Γ if the Christoffel symbols introduced in (3.2.7) were the coefficients

of such a connection. Which connection is the one described by the Christoffel symbols and how is it defined? The answer is: the Levi Civita connection. Its definition follows in the next paragraph.

Torsion and Torsionless Connections The notion of torsion was briefly anticipated in our historical outline. It applies only to affine connections and distinguishes them from general connections on generic fibre bundles. Intuitively torsion has to do with the fact that when we parallel transport vectors along a loop the transported vector can differ from the original one not only through a rotation but also through a displacement. While the infinitesimal rotation angle is related to the curvature tensor, the infinitesimal displacement is related to the torsion tensor. This was explicitly displayed in (3.2.12). Rigorously we have the following:

Definition 3.7.2 Let M be an m-dimensional manifold and denote an affine connection on its tangent bundle. The torsion T is a map:

T : X(M ) × X(M ) → X(M )

defined as follows:

X, Y X(M ) : T (X, Y) = −T (Y, X) ≡ XY − YX − [X, Y] X(M )

Given a basis of sections of the tangent bundle {−→ } we can calculate their com- e μ

mutators:

where the point dependent coefficients Kμν ρ (p) are named the contorsion coefficients. They do not form a tensor, since they depend on the choice of basis. For instance in the holonomic basis eμ = ∂μ the contorsion coefficients are zero, while they do not vanish in other bases. Notwithstanding their non-tensorial character they can be calculated in any basis and once this is done we obtain a true tensor, namely the torsion from Definition 3.7.2. Explicitly we have:

T (eμ, eν ) = Tμνρ eρ

(3.7.11)

Tμνρ = Γμν ρ − Γνμ ρ − Kμν ρ

Definition 3.7.3 An affine connection is named torsionless if its torsion tensor vanishes identically, namely if T (X, Y) = 0, X, Y X(M )

It follows from (3.7.12) that the coefficients of a torsionless affine connection are symmetric in the lower indices in the holonomic basis. Indeed if the contorsion vanishes, imposing zero torsion reduces to the condition:

The Levi Civita Metric Connection Consider now the case where the manifold M is endowed with a metric g. Independently from the signature of the latter (Riemannian or pseudo-Riemannian) we can define a unique affine connection which preserves the scalar products defined by g and is torsionless. That affine connection is the Levi Civita connection. Explicitly we have the following

Definition 3.7.4 Let (M , g) be a (pseudo-)Riemannian manifold. The associated Levi Civita connection g is that unique affine connection which satisfies the following two conditions:

(i)g is torsionless, namely T g (, ) = 0,

(ii)The metric is covariantly constant under the transport defined by g , that is:

Z, X, Y X(M ) : Zg(X, Y) = g( Zg X, Y) + g(X, Zg Y).

The idea behind such a definition is very simple and intuitive. Consider two vector fields X, Y. We can measure their scalar product and hence the angle they form by evaluating g(X, Y). Consider now a third vector field Z and let us parallel transport the previously given vectors in the direction defined by Z. For an infinitesimal displacement we have X → X + ZX and Y → Y + ZY. We can compare the scalar product of the parallel transported vectors with that of the original ones. Imposing the second condition listed in Definition 3.7.4 corresponds to stating that the scalar product of the parallel transported vectors should just be the increment along Z of the scalar product of the original ones. This is the very intuitive notion of parallelism.

It is now very easy to verify that the Christoffel symbols, defined in (3.2.7) are just the coefficients of the Levi Civita connection in the holonomic basis eμ = ∂μ ≡ ∂/∂xμ. As we already remarked, in this case the contorsion vanishes and a torsionsless connection has symmetric coefficients according to (3.7.12). On the other hand the second condition of Definition 3.7.4 translates into:

which admits the Christoffel symbols (3.2.7) as unique solution. There is a standard trick to see this and solve (3.7.13) for Γ . Just write three copies of the same equation with cyclically permuted indices:

Next sum (3.7.14) with (3.7.15) and subtract (3.7.16). In the result of this linear combination use the symmetry of Γλμ σ in its lower indices. With this procedure

you will obtain that Γλμ σ is equal to the Christoffel symbols.