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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$\begingroup$ 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! $\endgroup$– drbobmeisterCommented Feb 23, 2011 at 7:33
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$\begingroup$ Damn! Did it again. Very sorry! Answer almost ready. $\endgroup$– drbobmeisterCommented Jun 9, 2011 at 23:36
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$\begingroup$ I was just very tired. $\endgroup$– drbobmeisterCommented Jun 9, 2011 at 23:41
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2 Answers
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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.
answered Feb 8, 2011 at 4:16
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The differential of the Gauss map is the 2nd fundamental form.
answered Mar 9, 2012 at 15:33