Theorem Egregium: Intrinsic Extrinsic Gauss Curvature of 3D Surfaces

  • Thread starter wofsy
  • Start date
  • Tags
    Extrinsic
In summary, the product of the principal curvatures of a hypersurface in higher dimensions is the determinant of the Gauss map, and can be expressed in tensor language as the curvature 2-forms with respect to a principal frame field. This product is an intrinsic quantity known as the curvature tensor.
  • #1
wofsy
726
0
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?
 
Physics news on Phys.org
  • #2
Do you read curvature tensor ?
 
  • #3
Do you read curvature tensor ?

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

FAQ: Theorem Egregium: Intrinsic Extrinsic Gauss Curvature of 3D Surfaces

What is Theorem Egregium?

Theorem Egregium, also known as Gauss's Theorem, is a mathematical theorem that states the intrinsic and extrinsic curvatures of a 3D surface are independent of each other.

Who discovered Theorem Egregium?

The theorem was discovered by German mathematician Carl Friedrich Gauss in 1827.

What is the significance of Theorem Egregium?

Theorem Egregium has significant implications in differential geometry and the study of curved surfaces. It allows for the calculation of curvature of a surface without needing to refer to an external space.

What is the difference between intrinsic and extrinsic curvature?

Intrinsic curvature refers to the curvature of a surface as it exists within its own space, without any reference to an external space. Extrinsic curvature, on the other hand, refers to the curvature of a surface when it is embedded in a higher dimensional space.

How is Theorem Egregium used in real-world applications?

Theorem Egregium is used in various fields such as computer graphics, physics, and engineering to study and analyze curved surfaces. It is also used in differential geometry to study the intrinsic properties of surfaces without relying on external references.

Similar threads

Back
Top