Are both fundamental forms fundamental?

[Quetzalcoatl]

I've been studying some geometry and I've got a basic question:

 

The Riemann Curvature Tensor can be written using either the first fundamental form ([math]g_{\alpha\beta}[/math]) or the second fundamental form ([math]b_{\alpha\beta}[/math]), as indicated by the Codazzi-Peterson and Gauss equations. So I was wondering, is there a difference in the geometric information stored in both forms, or are they completely different, i.e. given one of these descriptions of a geometry (say [math]b_{\alpha\beta}[/math]), is it possible to find the other (say [math]g_{\alpha\beta}[/math])?

 

If not, what is the added value of each of the fundamental forms over the other?