Theorem Egregium: Intrinsic Extrinsic Gauss Curvature of 3D Surfaces

[wofsy]

The product of the principal curvatures of a surface in Euclidean 3 space, though defined extrinsically, is actually an intrinsic quantity, the Gauss curvature. This is the Theorem Egregium.

What about the product of the principal curvatures for higher dimensional hypersurfaces?

 

[hanskuo]

Do you read curvature tensor ?

 

[wofsy]

Do you read curvature tensor ?

hanskuo I am not sure what your question is. Explain.

 

[hanskuo]

In difernetial geometry, curvature is a tensor in higher dimensions.(more than 3 dimensions)
This is what I said curvature tensor.

 

[wofsy]

I am not referring to the curvature tensor but to the product of the principal curvatures of an embedded hypersurface. This product is the same as the determinant of the Gauss map.

 

[wofsy]

hanskuo, I apologize. I guess I lied a little. If one has already found principal directions on a hypersurface then the curvature 2-forms with respect to a principal frame field are the pairwise products of the principal curvatures times the wedge products of the dual 1-forms. In tensor language, if Ei are the principal directions then the curvature 2 forms are
R(X,Y,Ei,Ej) and the relevant equation is

R(X,Y,Ei,Ej) = KiKjEi*^Ej* where Ki is the i'th principal curvature and Ei* is the i'th dual i-form.