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Jan30-07, 11:10 AM
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mathwonk's Avatar
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i do not mean ambiently embedded surfaces are better, just easier to understand. Note the key point is that the tangent vectors in euclidean space can all be translated to the origin, i.e,. "the tangent bundle of R^n is 'trivial'".

so all one needs for a gauss map is to be embedded in a manifold with trivial tangent bundle. and every manifold does embed in euclidean space. moreover complex and real tori also have trivial tangent bundles, so all smooth subvarieties of tori also have gauss maps, and these were used crucially in the study of curves in their jacobians, by andreotti and mayer.

Even in manifolds without trivial tangent bundles, gauss maps, i.e. maps induced on tangent bundles, i.e. derivatives, have been used in study of moduli spaces and maps between them, especially by Carlson and Griffiths, and (earlier but less clearly) in my thesis.