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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2$\begingroup$ 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. $\endgroup$– Pedro Lauridsen RibeiroCommented Jan 8, 2013 at 14:58
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2 Answers
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It is a shame that this question is skipped in almost all modern textbooks! Except Nakahara I know of no other modern textbook.
There's a general contortion operation, sometimes called the Schouten braces in physicists' index calculus: $$T_{\{abc\}} := T_{abc} - T_{bca} + T_{cab}$$ See Schouten, p.132 formula (3.7). (This looks a bit unnatural/confusing to me: The last + should be a -.) (From this combined with index juggling and slot mishmash some of the deepest calculatorial confusions in theoretical physics have repeatedly arisen...).
The Schouten braces can also be applied to the nonmetricity tensor $\nabla g$:
Theorem: Let $\nabla$ be a (Koszul) covariant derivative with torsion $T$. Let $g$ be an arbitrary pseudo-Riemannian metric with Levi-Civita connection $\nabla^\circ$. There are 2 $g$-dependent contortion operators which embody Schouten braces plus index juggling, $C_1^g$ and $C_2^g$, whose details depend on your favorite variant of total covariant Koszul derivative, such that $$\nabla=\nabla^\circ+\frac{1}{2}C_1^g\cdot\nabla g+C_2^g\cdot T$$ (Thus $C_2^g\cdot T$ is +/- the contorsion, i.e. the contortion of torsion.)
The only serious reference I have for this not so hard theorem is J.A.Schouten, Ricci-Calculus 2nd ed., p.132 formula (3.5).
answered May 29, 2015 at 22:16
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$\begingroup$ This forgotten theorem is indeed due to Schouten and was a milestone in the development of general tensor calculus. Schouten, J. A.: 'Über die verschiedenen Arten der Übertragung in einer n- dimensionalen Mannigfaltigkeit, die einer Differentialgeometrie zugrundegelegt werden kann.' Mathematische Zeitschrift 13 (1922), 56-81. $\endgroup$ Commented Oct 17, 2016 at 15:52
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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.
answered Jan 8, 2013 at 15:16
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$\begingroup$ 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. $\endgroup$ Commented Jan 8, 2013 at 15:52