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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?

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Sorry about deleting my answer so quickly but I pushed the preview button while the page was jumping around due to rendering and I missed! Full answer available soon I hope! – drbobmeister Feb 23 '11 at 7:33
Damn! Did it again. Very sorry! Answer almost ready. – drbobmeister Jun 9 '11 at 23:36
I was just very tired. – drbobmeister Jun 9 '11 at 23:41
up vote 4 down vote accepted

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.

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

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