I have heard that in a space with torsion autoparallel curves and geodesics are different and they coincide when the torsion vanishes. But I couldn't find any definition for the autoparallel curve. Please help.
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Let $M$ be a pseudo-Riemannian manifold endowed with an affine connection $\overline\nabla$, which is not necessarily the Levi-Civita connection $\nabla$.
Definition 1. A curve in $TM$ is called horizontal by $\ \overline\nabla$ if its points are parallel transports of each other.
Definition 2. A curve $c$ in $M$ is called autoparallel by $\overline\nabla$ if its natural lift to $TM$ is horizontal by $\overline\nabla$. Natural lift means a curve in $TM$ consisting of the tangent vectors of $c$.
Definition 3. A curve in $M$ is called geodesic if it is autoparallel by the Levi-Civita connection $\nabla$.
According to Spivak$^1$, if $\overline\nabla$ differs from $\nabla$ in a torsion, i.e. if there is a skew-symmetric tensor field $\overline T$ of type $\binom{2}{0}$ so that $\overline\nabla_XY=\nabla_XY+\overline T(X,Y)$, then the autoparallels of $\overline\nabla$ and the geodesics are the same, otherwise not. Note that the torsion of $\overline\nabla$ is then $2\overline T$.
$^1$Michael Spivak: A Comprehensive Introduction to Differential Geometry Volume 2, p.250, 16. Corollary. (see here)
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