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Maybe this explains, that space is twisting along geodesics if the torsion is non-zero. … So torsion can be viewed either as a property of the soldering form (choose it better if you want to get rid of torsion), or as a property of $\nabla$ (if you identify $TM$ with $E$ with the given soldering …

Torsion is measuring something different: It is the covariant derivative of the soldering form $\sigma\in\Omega^1(M,E)$ which you use to identify the vector bundle $E$ with $TM$, where $E$ is the bundle …

You can compute torsion either on $P$ or on $M$ and they correspond to each other. Edit: Parallel transport and torsion. … The only way to get torsion is using again 24.2: $\nabla_XY = \partial_t|_0 Pf(Fl^X,t)^* Y$ and building torsion out of this. …

Edit: Answering Robert, moved to 3-space to give a general example: A simple example in $M=\mathbb R^3$: Let $N=0\times \mathbb R^2$ and put $$ \nabla_XY = dY(X) + \begin{pmatrix}X^T\,A^1\,Y \\ X^T\ …

Levi-Civita means metric compatible and torsion free. Adding a skew symmetric $\binom{1}{2}$ tensor field (= your favorite torsion) to a covariant derivative does not change metric compatibility. …