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Let $M$ be a Riemannian manifold with affine connection such that the metric is covariantly constant (so that the connection equals the Levi-Civita connection up to torsion).

I know the interpretation of torsion and curvature in terms of rolling without slipping (that is its interpretation as the curvature of an underlying Cartan connection). What I am looking for is an interpretation in terms of geodesics (that means free-falling particles in Einstein-Cartan theory). As it is well known, the family of all geodesics does not depend on the torsion (two connections that are the same up to torsion have the same geodesics), this interpretation also has to use the concept of parallel displacement directly. For example, one could talk about geodesics starting parallel, etc.

In case of vanishing torsion, the Jacobi equation for Jacobi fields (that is infinitesimal families of parallel geodesics) give me a complete description (and interpretation) for the curvature tensor as a relative acceleration of nearby geodesics. In case of non-vanishing torsion, the equation becomes more complicated as a covariant derivative of the torsion enters as well.

Is there a similar (probably first-order) equation for geodesics in which the torsion enters directly and gives me a direct interpretation? Can the Jacobi equation (or the underlying problem) be reformulated so that it stays the same independent of the torsion?

I have read: http://en.wikipedia.org/wiki/Torsion_%28differential_geometry%29#Twisting_of_reference_frames but I have difficulties to interpret this result. And it is lacking any reference.

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1 Answer 1

The geodesic structure

  • $geo: TM\times \mathbb R\supset U \to M$ with
    $geo(X_x,0)=x$,
    $\partial_t|_0 geo(X_x,t) = X_x$
    $geo(geo(X_x)'(s),t)=geo(X_x,s+t)$
    $U\cap(\lbrace X_x\rbrace\times \mathbb R) = \lbrace X_x\rbrace\times \text{intervall}$

determines the connection up to a skew $\binom12$-tensor field, and the torsion free connection can be recontructed from the geodesic spray (a vector field on $TM$). See 22.6 ff in here. If $\nabla$ is torsion-free, and $T:TM\times_M TM \to TM$ is a skew tensor field, then $\nabla'_XY:=\nabla_XY+T(X,Y)$ has the same geodesics and torsion $2T$. Compute the curvature $R'$ of $\nabla'$ in terms of the curvature $R$ of $\nabla$ and $T$ and write the Jacobi equation for $\nabla'$ (including the torsion) and expand it in terms $\nabla$ and $T$. It is the same equation as the Jacobi equation for $\nabla$; all the extra terms cancel.

In this sense there is no way to see the torsion just from the geodesics alone.

See also this question and its answers.

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That the geodesics (or the geodesic spray) only encode the affine connection up to torsion, that I know. But I don't think that it means that my question cannot be answered: even in the case of the Jacobi equation, which is on geodesic variation, the torsion enters. This is, of course, because the Jacobi equation talks about the acceleration of the distance between geodesics and the covariant derivative enters in this acceleration directly. I'll edit my question above to make it more clearly. –  Marc Nieper-Wißkirchen Oct 20 '13 at 8:28
    
Maybe it is not what you wanted; but in the Jacobi equation the torsion term enters only in order to cancel the torsions entering via the connection and its curvature. By the way, see also the paper: Peter W. Michor: The Jacobi Flow. Rend. Sem. Mat. Univ. Pol. Torino 54, 4 (1996), 365-372 (available via my homepage). –  Peter Michor Oct 20 '13 at 9:12

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