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In GR we focus on the Levi-Civita connection for parallel transport reasons and preserving the inner product. We are usually not formally taught other types of connections (aside from maybe the Weitzenblock connection or the Einstein Cartan connection with torsion).

Out of pure mathematical curiosity without connection to the real world, what would be an example of an affine connection that does not preserve the inner product (which releases the metric compatibility condition that we impose in GR)? I know this is very general, but are there any specific examples that are used in other areas of math?

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  • $\begingroup$ Does the affine connection on a Mobius strip preserve the inner product or is there a sign change? $\endgroup$
    – FlatterMann
    Commented Oct 9, 2022 at 3:41
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    $\begingroup$ Take any $(2,1)$ tensor $T$. If $\nabla$ is your previous connection, then $\nabla +T$ is also a connection but it will not preserve the inner product for generic $T$. $\endgroup$
    – Didier
    Commented Oct 9, 2022 at 16:18
  • $\begingroup$ @Didier Thanks! just to be sure, is $\nabla$ the affine connection or the connection coefficients? $\endgroup$
    – B K
    Commented Oct 12, 2022 at 8:52
  • $\begingroup$ It is the connection. Its coefficients are given by $\nabla_{\partial_i}\partial_j = \Gamma^k_{ij}\partial_k$. $\endgroup$
    – Didier
    Commented Oct 12, 2022 at 8:53

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