Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Intrinsic extrinsic

  1. Apr 7, 2008 #1
    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?
  2. jcsd
  3. Apr 8, 2008 #2
    Do you read curvature tensor ?
  4. Apr 8, 2008 #3
    Do you read curvature tensor ?

    hanskuo I am not sure what your question is. Explain.
  5. Apr 8, 2008 #4
    In difernetial geometry, curvature is a tensor in higher dimensions.(more than 3 dimensions)
    This is what I said curvature tensor.
  6. Apr 8, 2008 #5
    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.
  7. Apr 8, 2008 #6
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Intrinsic extrinsic Date
Manifolds: extrinsic and intrinsic Sep 16, 2015
Conversions between intrinsic/extrinsic coordinates Feb 26, 2014
Extrinsic intrinsic Feb 8, 2010
Extrinsic and Intrinsic Curvature Mar 17, 2005
Extrinsic/Intrinsic curvature Jun 9, 2004