# Difference tensor between two connections

OP's question is probably spurred by the fact that Ref. 1 forgets to mention that:

1. The other connection $$\nabla:\Gamma(TM)\times \Gamma(TM)\to \Gamma(TM)$$ with lower Christoffel symbols $$\Gamma_{ij,k}~:=~ \Gamma_{ij}^{\ell}~g_{\ell k}\tag{1}$$ is still assumed to be compatible with the metric $$\nabla g = 0\qquad\Leftrightarrow\qquad \partial_kg_{ij}=\Gamma_{ki,j}+\Gamma_{kj,i}.\tag{2}$$

2. The contorsion tensor $$K~\in~\Gamma\left(T^{\ast}M\otimes \bigwedge^2(T^{\ast}M)\right)$$ is defined$$^1$$ as the difference between the metric-compatible $$\nabla$$ and the Levi-Civita connection $$\nabla^{LC}$$, i.e. $$K_{i,jk}~:=~\Gamma_{ij,k}-\Gamma^{LC}_{ij,k} ~\stackrel{(2)}{=}~-K_{i,kj}.\tag{3}$$

3. The torsion tensor $$T~\in~\Gamma\left(\bigwedge^2(T^{\ast}M)\otimes T^{\ast}M \right)$$ is defined$$^1$$ as $$T(X,Y)~:=~\nabla_XY-\nabla_YX-[X,Y]$$ $$\quad\Leftrightarrow\quad T_{ij,k}~:=~\Gamma_{ij,k}-\Gamma_{ji,k}~\stackrel{(3)}{=}~K_{i,jk}-K_{j,ik}.\tag{4}$$

4. One may show that the inverse relation to (4) is $$K_{i,jk}~\stackrel{(4)}{=}~ \frac{1}{2}\left(T_{ij,k}-T_{jk,i}+T_{ki,j}\right),\tag{5}$$ and vice-versa.

5. Let us decompose the contorsion tensor $$K_{i,jk} ~=~ \frac{1}{2}(K^+_{i,jk}+K^-_{i,jk}), \tag{7}$$ in components $$K^{\pm}_{i,jk}~:=~K_{i,jk}\pm K_{j,ik},$$ $$K^+_{i,jk}~=~T_{ki,j}+T_{kj,i}, \qquad K^-_{i,jk}~=~T_{ij,k}, \tag{8}$$ that are symmetric/antisymmetric wrt. the first two indices $$i\leftrightarrow j$$.

6. The geodesic equation reads $$0~=~\nabla^{LC}_{\dot{\gamma}}\dot{\gamma} \qquad\Leftrightarrow\qquad \ddot{\gamma}^{\ell}~=~-\Gamma^{LC,\ell}_{ij}\dot{\gamma}^i\dot{\gamma}^j$$ $$\qquad\Leftrightarrow\qquad -g_{k\ell}\ddot{\gamma}^{\ell}~=~\Gamma^{LC}_{ij,k}\dot{\gamma}^i\dot{\gamma}^j.\tag{9}$$

7. In contrast, the auto-parallel equation $$0~=~\nabla_{\dot{\gamma}}\dot{\gamma} \qquad\Leftrightarrow\qquad \ddot{\gamma}^{\ell}~=~-\Gamma^{\ell}_{ij}\dot{\gamma}^i\dot{\gamma}^j$$ $$\qquad\Leftrightarrow\qquad -g_{k\ell}\ddot{\gamma}^{\ell}~=~\Gamma_{ij,k}\dot{\gamma}^i\dot{\gamma}^j~=~\left(\Gamma^{LC}_{ij,k}+\frac{1}{2}K^+_{i,jk}\right)\dot{\gamma}^i\dot{\gamma}^j\tag{10}$$ can detect the symmetric part $$K^+_{i,jk}$$ of the contorsion.

8. Observation: $$T_{ij,k} \text{ is totally antisymmetric} \qquad\Leftrightarrow\qquad K_{i,jk} \text{ is totally antisymmetric}$$ $$\qquad\Leftrightarrow\qquad K^+_{i,jk}~=~0.\tag{11}$$ In the affirmative case, we have $$T_{ij,k}=2K_{i,jk}$$, and the geodesic & auto-parallel equations (9) & (10) are the same.

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$$^1$$ Pertinent applications of the musical isomorphism are implicitly implied from now on.