Understanding GR Better: Metric Spaces & Differential Geometry Courses

[madness]

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?

 

[atyy]

The "first fundamental form" is the "line element".

 

[madness]

The "first fundamental form" is the "line element".

By line element you mean the infinitessimal interval right? Anyway, when calculating the curvature of a surface, you consider the first and fundamental forms as matrices and find the eigenvalues of II with respect to II, i.e. det(II-kI)=0. Then the matrix of I is the metric in GR, which is very different from the usual metric studies in metric spaces.