# 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},
\end{aligned}

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
MetricFixed0
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.