So I've taken courses in metric spaces and differential geometry this year and one of the main reasons for choosing these courses was to understand GR better. I'm finally getting round to actually learning my differential geometry course, and have come across the "first fundamental form" of a surface. Am I right in thinking that this is the "metric" of GR (more specifically it is the interval, and the metric is the matrix of the first fundamental form in the right basis)? Does the second fundamental form describe curvature in GR? Is it related to the Riemann tensor?