Independent quantities and dependencies

While in general the curvature on a Riemannian manifold does not determine the metric, for a manifold with connection that is compact, simply connected, and has no regions of constant curvature (i.e. there is no way to “stretch” the manifold without affecting the curvature), knowledge of the curvature at all points determines the connection (up to changes in frame), and therefore the metric that makes this connection Levi-Civita (up to a constant scaling factor).

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