Independent quantities and dependencies

From their definitions, the parallel transport and connection in general determine each other. It can be shown that every manifold admits a connection, and every other connection can be obtained by adding a frame-independent \({gl\left(\mathbb{R}^{n}\right)}\)-valued 1-form (tensor field of type \({(1,2)}\)) to it. A connection with \({\mathrm{Hol}(M)\subseteq O(r,s)}\) may not be able to be defined on a manifold, but if it can, then given this connection (apart from special cases) it uniquely determines a compatible signature \({(r,s)}\) metric (up to constant scaling factors). The connection can then be written as the sum of the Levi-Civita connection \({\overline{\Gamma}}\) of this metric and a tensor \({K}\) called the contorsion tensor (AKA contortion tensor), which using previous relations can be expressed in terms of the torsion:

\begin{aligned}\Gamma&=\overline{\Gamma}+K\\K_{abc}&=\frac{1}{2}\left(T_{bac}+T_{cab}-T_{abc}\right)\\\Rightarrow T_{abc}&=K_{acb}-K_{abc}\end{aligned}

It is important to note that the definition of contorsion incorporates the metric via the lowered index; thus for example changing \({\overline{\Gamma}}\) (and thus the metric) while holding \({K}\) constant alters \({T}\). Also note that \({K}\) is anti-symmetric in its first two indices since \({T}\) is anti-symmetric in its last two. The geodesic equation is

\begin{aligned}\left(\nabla_{v}v\right)^{a} & =\left(\overline{\nabla}_{v}v\right)^{a}+K^{a}{}_{bc}v^{b}v^{c}\\
& =\left(\overline{\nabla}_{v}v\right)^{a}-T_{bc}{}^{a}v^{b}v^{c},

so that the autoparallel geodesics of a connection with non-zero torsion coincide with the Riemannian geodesics of the Levi-Civita connection iff \({K}\) (and therefore \({T}\)) is completely anti-symmetric. This can only occur in at least three dimensions, where in an orthonormal frame the anti-symmetry indicates that parallel transport “spins” as in the last figure in the section on torsion.

Δ If one uses the convention which reverses the lower indices of the connection coefficients, the same happens to the contorsion tensor to yield \({K_{abc}=\frac{1}{2}\left(T_{bac}+T_{cab}+T_{abc}\right)}\). In the literature one finds authors who use our convention for the lower indices of the connection coefficients, but reverse the sign of the torsion tensor and/or halve it, which sign reverses and/or doubles our expression for the contorsion tensor.
Δ Given a connection \({\Gamma^{\lambda}{}_{\mu\sigma}}\) in a coordinate frame with associated metric \({g}\), it is important to remember that while the anti-symmetric part of \({\Gamma}\) is proportional to the torsion, the symmetric part is not the Levi-Civita connection for \({g}\) (although it is for some other metric, and has the same autoparallel geodesics as \({\Gamma}\)). In particular, one sometimes sees statements to the effect that since only the symmetric part of \({\Gamma}\) affects the geodesic equation, “torsion does not affect geodesics.” What is true is that subtracting the torsion from \({\Gamma}\) leaves autoparallel geodesics unchanged (but does change the metric and hence Riemannian geodesics). But it is important to remember that the more geometric notion of modifying the torsion of \({\Gamma}\) (leaving \({\overline{\Gamma}}\) and thus \({g}\) unchanged) adds a symmetric part to \({\Gamma}\) which does in fact change geodesics.

If the curvature is given over \({M}\), there is at most one metric (also apart from special cases, up to a scaling factor, and for \({n>2}\)) whose Levi-Civita connection yields this curvature.

If we choose coordinate charts and use coordinate frames on \({M^{n}}\), we can calculate the number of independent functions and equations associated with the various quantities and relations we have covered, and use them to verify the associated dependencies.

Quantity / relationViewpointCount
MetricSymmetric matrix of functions\({n(n+1)/2}\)
Coordinate frameFixed0
Connection\({gl(n,\mathbb{R})}\)-valued 1-form\({n^{3}}\)
Metric conditionDerivative of metric\({n^{2}(n+1)/2}\)
Torsion-free condition\({\mathbb{R}^{n}}\)-valued 2-form\({n^{2}(n-1)/2}\)

The choice of coordinates determines the frame, leaving the geometry of the Riemannian manifold defined by the \({n(n+1)/2}\) functions of the metric. A torsion-free connection consists of \({n^{3}-n^{2}(n-1)/2=n^{2}(n+1)/2}\) functions. The metric condition is exactly this number of equations, allowing us in general to solve for the connection if the metric is known, or vice-versa (up to a constant scaling factor).

Alternatively, we can look at things in a orthonormal frame:

Quantity / relationViewpointCount
Orthonormal frame\({n}\) vector fields\({n^{2}}\)
Change of orthonormal frame\({SO(n)}\)-valued 0-form\({n(n-1)/2}\)
Connection\({so(n)}\)-valued 1-form\({n^{2}(n-1)/2}\)
Metric conditionAutomatically satisfied0
Torsion-free condition\({\mathbb{R}^{n}}\)-valued 2-form\({n^{2}(n-1)/2}\)

Here the metric is fixed, determined by the frame, which consists of \({n^{2}}\) functions, but is determined only up to a change of orthonormal frame (rotation); this yields \({n^{2}-n(n-1)/2=n(n+1)/2}\) functions, consistent with the metric function count above. The torsion-free condition is the same number of equations as the connection has functions, so that in general the torsion-free connection can be determined by the orthonormal frame.

An Illustrated Handbook