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 Levi-Civita connection of this metric is then obtained by subtracting a torsion- and metric-dependent tensor \({K}\) called the contorsion tensor from the connection coefficients:

\begin{aligned}\Gamma_{\mathrm{Levi-Civita}} & =\Gamma-K\\
K_{abc} & =\frac{1}{2}\left(T_{bac}+T_{cab}-T_{abc}\right)
\end{aligned}

Note that \({K}\) is anti-symmetric in its first two indices since \({T}\) is anti-symmetric in its last two. Denoting the Levi-Civita covariant derivative \({\overline{\nabla}}\), we see that the geodesic condition

\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}

means that the geodesics of the connection coincide with those of the Levi-Civita connection iff \({K}\) and therefore \({T}\) are both completely anti-symmetric, i.e. \({3}\)-forms.

Δ If one uses the convention which reverses the lower indices of the connection coefficients, the contorsion tensor is instead \({K_{abc} = \frac{1}{2}\left(T_{abc}+T_{cab}+T_{bac}\right)}\). 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, which reverses the sign of our expression for the contorsion tensor.

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.

An Illustrated Handbook