Given a surface M in Euclidean space, we have the generalized-Gauss-map 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.