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Do we have a choice in defining the covariant derivative by the use of a set of coefficient functions(Christoffel gammas)? If so, could we then say that these coefficient functions need not to coincide with the ones from the equation obtained from variational principle where we are looking for the shortest curve?

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  • $\begingroup$ A pseudo-Riemannian manifold $(M,g)$ has a canonical connection, namely the Levi-Civita connection. But that's just one connection out of many possible connections. Possible duplicate: physics.stackexchange.com/q/342821/2451 $\endgroup$ – Qmechanic Dec 30 '18 at 10:33