Given a surface M in Euclidean space, we have the generalizedGaussmap G, i.e. map the tangent spaces into the Grassmannian G(2,n). What is the relation between DG and the second fundamental form of M, and the Gauss curvature?

Everything generalizes nicely. A nice approach to working this out is described in Griffiths, P. On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41 (1974), 775–814. 


The differential of the Gauss map is the 2nd fundamental form. 

