Shape of Spacetime: Curvature Tensor or Metric?

[jfy4]

hello,

quick question, If i am interested in the shape of space time, am i interested in the curvature tensor, or the metric?

Thanks

 

[bcrowell]

quick question, If i am interested in the shape of space time, am i interested in the curvature tensor, or the metric?

The curvature tensor is more relevant. E.g., you can have a metric that looks very complicated, but is actually a flat-space metric expressed in funky coordinates. In this situation, the curvature tensor would vanish, and knowing that would be more informative than just staring at the metric.