Killing vectors in space with torsion?


I was wondering whether it is possible or not to define Killing vectors in a manifold with torsion. As a first approach I was reasoning like:

Taking as definition of killing vectors, a vector field that: any set of points when displaced along a Killing integral lines, by equal amounts will have the distances between the points kept unchanged. This implies that the metric is unchanged along the Killing directions.

In a space endowed with torsion, I think it is perfectly possible to define vector fields that leave the metric unchanged. But I'm not so sure these vectors can be used to displace points in the sense previously explained, with distances kept constant. Even though the metric is kept unchanged I'm wandering whether the existence of torsion might change distances between points.

If the entire manifold is displaced along vectors of previous paragraph, the new manifold cannot be considered as an isometrie, because altough metric field is invariant the torsion field in general will not be, since no special condition on it is imposed by definition Killing vectors. Or is this wrong?

Thanks advance,


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The entire concept of torsion is associated with affine connections while the concept of a Killing field is related to the Lie derivative of the metric. Those two concepts are not directly related. You can have a Killing field without even defining an affine connection and it is perfectly possible to define a Killing field in the presence of a connection with non-zero torsion.

There is another issue with the metric and affine connections, which is whether or not the connection is metric compatible. If it is then parallel transport will preserve lengths and angles between the parallel transported vectors. It is also perfectly possible to have a metric compatible connection with non-zero torsion. However, there is only one torsion free connection that is metric compatible - the Levi-Civita connection.

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