If already known the Action and unkown the Metric, how to get geodesic equation?

yicong2011

If already known the form of Action and unkown the Metric, how to derive the geodesic equation?

yicong2011

Another question, if already known the Action (unknown the metric), how to derive Riemann Curvature from the Action?

atyy

http://arxiv.org/abs/1102.0529, section 19.1 derives the geodesic equation from the action