Theorema Egregium: Does Gauss's Theorem Generalize?

[lavinia]

Does Gauss's theorem that the Gauss curvature - as computed from the determinant of the differential of the Gauss mapping - is intrinsic, generalize to a hypersurface of a higher dimensional Euclidean space?

 

[mathwonk]

there is a generalization on page 78 of hicks' notes on differential geometry, van nostrand math studies #3, 1965. but there seems to be something special about the case of hypersurfaces of 3 manifolds. see also p. 29.

http://www.wisdom.weizmann.ac.il/~yakov/scanlib/hicks.pdf

 

[lavinia]

there is a generalization on page 78 of hicks' notes on differential geometry, van nostrand math studies #3, 1965. but there seems to be something special about the case of hypersurfaces of 3 manifolds. see also p. 29.

http://www.wisdom.weizmann.ac.il/~yakov/scanlib/hicks.pdf

Thanks Mathwonk. So the Theorem Egregium generalizes to a relationship between the sectional curvatures of a tangent 2 dimensional hyperplane of a hypersurface of a Riemannian manifold. For a surface in three space this says that the sectional curvature is the determinant of the Gauss map. This is a local isometry invariant of the surface because the 2 dimensional hyperplane is unique for a surface.

In higher dimensions it is not unique.

But for an even dimensional hypersurface there are independent 2 planes that are spanned by pairs of principal vectors (directions of principlal curvature). The product of their sectional curvatures is the determinant of the Gauss map. Hmmm.. maybe this is not intrinsic.