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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,