TITLE: Generalized Gauss map
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: The differential of the Gauss map is the 2nd fundamental form.