Intrinsic extrinsic

  • Thread starter wofsy
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Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
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Do you read curvature tensor ?
 
  • #3
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Do you read curvature tensor ?

hanskuo I am not sure what your question is. Explain.
 
  • #4
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In difernetial geometry, curvature is a tensor in higher dimensions.(more than 3 dimensions)
This is what I said curvature tensor.
 
  • #5
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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.
 
  • #6
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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.
 

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