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