A question about metric compatibility equation

[renormalize]

I think it perhaps a conbination of rotation invariance of metric and curvature determine the property of path dependence of metric.

It's frustrating when you continue to make these statements about what you "think". What matters is what you can prove! The whole point of my post #46 is to demonstrate (prove!) that the metric $g$ is unchanged by parallel-transport through an infinitesimal-distance along a path by any metric-compatible connection $\Gamma$ (see eq.(10)), regardless of the curvature or torsion. So I must cease commenting in this thread unless you do me the courtesy of one of the following:

• Prove using equations (not words) that "curvature determine the property of path dependence of metric", or
• Prove using equations (not words) that there is an error in post #46 that renders eq.(10) invalid, or
• Acknowledge that eq.(10) is valid and proves that $g$ is invariant when parallel-transported by a metric-compatible $\Gamma$.

 

[berkeman]

Thread closed for Moderation...

 

[PeterDonis]

After moderator review, this thread will remain closed as the OP question has been sufficiently addressed.