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Is there any theorem which states any general linear connection can be decomposed into another linear connection plus the contortion tensor ? i didn't find any References

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I believe that would be Proposition 7.9, page 146 of the book of S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry - Volume I" (Wiley, 1963). Moreover, the second connection has zero torsion and the same geodesics as the first. Notice, however, that such a question is not really research-level and hence more suitable for math.stackexchange. –  Pedro Lauridsen Ribeiro Jan 8 '13 at 14:58

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Check out Kobayashi and Nomizu's Foundations of Differential Geometry, Volume 1. On page 159, when proving the existence of the Levi-Civita connection (Theorem IV.2.2), they pick an arbitrary metric connection and add the contorsion tensor to it and show that it is a metric connection with vanishing torsion. Hence any metric connection can be written as the difference of the Levi-Civita connection and its contorsion tensor.

Another reference is Section 7.2.6 of Nakahara's Geometry, Topology, and Physics. See equations (7.30)-(7.35) for Nakahara's derivation.

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Wordline's question concerns general linear connections, not just metric ones, I suppose. Proposition III.7.9 on page 146 (quoted in my comment above) does the job at the required level of generality, but the argument is essentially the same. –  Pedro Lauridsen Ribeiro Jan 8 '13 at 15:52

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