Talk:Gaussian curvature

message: What's the relationship between Gaussian curvature and Ricci scalar? --Itinerant1 20:19, 30 March 2007 (UTC)
replies: text: Have a look at Curvature of Riemannian manifolds, which explains a bit more. --Salix alba (talk) 22:18, 30 March 2007 (UTC)

message: It may be very obvious, but does anybody feel it should be made clear in the figure caption that the colormap does not represent Gaussian curvature?  — Preceding unsigned comment added by Dfsp spirit (talk • contribs) 08:13, 25 April 2019 (UTC)
thread_title: Figure containing hyperboloid, cylinder, and sphere

message: Am I the only one whom the "Paper" section strikes as super-pedantic and silly?67.172.93.9 06:15, 4 December 2007 (UTC)
thread_title: Paper

message: The formula in terms of the connection appearing at the beginning of the "alternative definitions" section may not be correct, at least if one defines e_1 and e_2 the usual way.  If their commutator is nonzero, aren't we missing a term here?  Tkuvho (talk) 09:27, 12 April 2010 (UTC)
thread_title:  error in alternative definition? 

message: There should be three sections in the article, each talking about negative, positive or zero Gaussian Curvature,  with different examples and what that means with respect to the principal curvatures and vectors. Basically, there should be more insightful discussion with Gaussian Curvature as it has very far reaching consequences in differential geometry.  —Preceding unsigned comment added by 128.100.86.103 (talk) 23:09, 19 April 2011 (UTC)
thread_title:  Better Classification 

message: The article has for the F(x,y,z)=0 
::K=\frac{[F_z(F_{xx}F_z-2F_xF_{xz})+F_x^2F_{zz}][F_z(F_{yy}F_z-2F_yF_{yz})+F_y^2F_{zz}]-[F_z(-F_xF_{yz}+F_{xy}F_z-F_{xz}F_y)+F_xF_yF_{zz}]^2}{F_z^2(F_x^2+F_y^2+F_z^2)^2}
With the comment "This one is too long, cannot figure out how to divide it nicely". The reference comes from Mathworld http://mathworld.wolfram.com/GaussianCurvature.html, which references a work by Trott. This is a little unsatisfactory as there is a F_z^2 factor on the bottom and ideally we should see symmetry in the three coordinates.
I've found a nicer reference  which presents a different form. 
K=\frac{ \nabla F\ H^{*}(F)\ \nabla F^{\mathsf T} }{ |\nabla F|^4 }
Where \nabla F is the gradient and H^{*}(F) is the adjoint of the Hessian. It also references Spivak who has perhaphs the nicest presentation
K=-\frac{ 
\begin{vmatrix}
H(F) & \nabla F^{\mathsf T} \\
\nabla F & 0 
\end{vmatrix}
}{ |\nabla F|^4 }
or explicitly
K=-\frac{ 
\begin{vmatrix}
F_{xx} & F_{xy} & F_{xz} & F_x \\
F_{xy} & F_{yy} & F_{yz} & F_x \\
F_{xz} & F_{yz} & F_{zz} & F_x \\
F_{x} & F_{y} & F_{z} & 0 \\
\end{vmatrix}
}{ |\nabla F|^4 }
I've checked the formula and they expand to the same thing, the F_z^2 does cancel nicely.--Salix alba (talk): 20:33, 14 April 2015 (UTC)
replies: text: Just wondering: Is that right-hand column of the matrix of the numerator determinant correct, with three Fx's? 2601:200:C082:2EA0:D8BD:7D54:9940:6DDD (talk) 23:39, 21 April 2023 (UTC)
thread_title:  Implicit form 

message: The second point in the 'informal definition' is very obscure.  It uses the phrase 'plane orthogonal to the surface' without explaining what this means.  If it is the same as a 'normal plane' (which has already been defined) it is unnecessary and potentially confusing, while if it is different from a normal plane, it is even more important to say what it is.  The passage then talks about rotating the orthogonal plane in two directions, without saying what they are - clockwise and anticlockwise? - and asserts that 'the normal curvatures will be zero'. I can see that if a value goes continuously from positive to negative, it must be zero somewhere, but the plural 'curvatures' implies more than one position of zero curvature, which is either wrong or not clearly explained.2A00:23C5:6487:4701:845C:797:3656:22D7 (talk) 11:59, 24 June 2020 (UTC)
thread_title:  Obscure passage 

message: The caption accompanying one illustration, depicting two surfaces, reads as follows:
"Two surfaces which both have constant positive Gaussian curvature but with either an open boundary or singular points."
One surface resembles the outer half of a torus of revolution, the shape of a wrist bangle, while the other resembles an (American) football. The two non-smooth points of the football must be the "singular points" referred to. But it is not clear what "open boundary" has to do with the bangle.
This section needs to give the metric to say what E, F, and G are, which are extensively used but never defined, and also say what L, M, and N are in the first formula.  — Preceding unsigned comment added by Donpage (talk • contribs) 15:51, 12 April 2024 (UTC)
thread_title:  Problematic caption