So as the title says, what are the physical reasons behind requiring the connection between tangent vector spaces to be metric-compatible?(adsbygoogle = window.adsbygoogle || []).push({});

My guess is that this is desired from wanting different points in space-time to be "equivalent", in the sense that if any two vectors at a point are the same at another point, their inner product is the same, meaning that if a physical quantity is the same on different tangent spaces, their properties are still the same (no way of distinguishing one point on the manifold from the other).

I'm not used to talk in proper mathematical language so i apologize if the argument is not properly presented.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Physical reasons for having a metric-compatible affine connection?

Have something to add?

Draft saved
Draft deleted

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**