Talk:Second fundamental form

Number or vector
Something here is wrong:


 * $$\langle R(u,v)w,z\rangle =\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle.$$


 * $$\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle.$$

The second fundamental form is a bilinear operator. II(u,v) is a number, not a vector. Why are we taking inner products of numbers?

68.1.154.81 (talk) 21:11, 13 March 2008 (UTC)


 * Thanks for catching that. At this point in the text, the second fundamental form refers to a normal-bundle valued bilinear form (i.e., it's vector valued, not scalar valued).  I have added a section break to set up the contrast a little better, but it probably needs still more clarification.  silly rabbit  (  talk  ) 21:33, 13 March 2008 (UTC)


 * There is a further mistake due to that definition. It is stated that the second fundamental form is well defined up to a sign, but this is only true for the scalar valued one. The second fundamentalform defined like it is here has no sign ambiguity. — Preceding unsigned comment added by 130.60.188.204 (talk) 10:12, 20 June 2014 (UTC)

C.f. Hessian?
It seems that the second fundamental form is related to the Hessian matrix in that both define the local curvature of a surface. Obviously there are caveats like that a Hessian is usually describing the curvature with respect to the z direction of a surface over Rn where the slope may be nonzero whereas II, being in the tangent space of a manifold never has to worry about slope. Is there a connection? Is II a generalization of a Hessian to a manifold in some sense? —Ben FrantzDale (talk) 12:42, 10 August 2010 (UTC)

What is this notation?
Okay, I know every field has slightly different notation for linear algebra, but seriously, what is with this equation?


 * $$b_{\alpha \beta} = r_{,\alpha \beta}^{\ \ \,\gamma} n_{\gamma}\,. $$

There is a comma in there. A comma! And is there a meaning to the periods at the end of the equations in that section? And what's with the gammas? — Preceding unsigned comment added by Joeedh (talk • contribs) 07:00, 16 February 2020 (UTC)


 * The gamma might be Einstein summation convention where summation sign is dropped. r is Some kind of tensor, the comma might be used to separate the covariant and contravariant indices. Although there seems to be too many indices here. Knowing the context might help. --Salix alba (talk): 20:30, 16 February 2020 (UTC)


 * This appears only once in the article and is probably a typographical error. Evidently the comma should be preceded by a backslash.  With the backslash reinserted it displays like this and makes perfect sense:


 * $$b_{\alpha \beta} = r_{\,\alpha \beta}^{\ \ \,\gamma} n_{\gamma}\,. $$


 * —Anita5192 (talk) 22:19, 16 February 2020 (UTC)
 * The comma is indeed a physicist's notation for partial derivative, as described in the text. You can see examples of this at Covariant derivative, at https://en.wikibooks.org/wiki/General_Relativity/Coordinate_systems_and_the_comma_derivative, by researching the "comma derivative", or consulting any physics text that deals extensively with tensors (especially texts on general relativity).
 * Thus the two lower indices, following the comma, indicate the directions of the two partial derivatives. The upper index is nothing special; it indexes into $R^{3}$, the codomain of $r$. It does not make sense to add additional indices to $r$ without indicating a derivative.
 * I will restore the comma.
 * (This tensor notation clashes slightly with the vector notation used at the top of this section, but I'll leave that as is for now.) JLLong (talk) 20:33, 18 January 2024 (UTC)

Confusion between quadratic form and symmetric bilinear form
The section Generalization to arbitrary codimension contains the following passage:

"The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined by
 * $$\mathrm{I\!I}(v,w)=(\nabla_v w)^\bot\,, $$

where $$(\nabla_v w)^\bot$$ denotes the orthogonal projection of covariant derivative $$\nabla_v w$$ onto the normal bundle."

No question, there is a one-to-one correspondence between symmetric bilinear forms on the one hand and quadratic forms on the other hand (at least in characteristic zero).

But regardless of that, the two concepts are not the same. A quadratic form is a certain kind of function of *one* element of a vector space; a symmetric bilinear form is a certain kind of function of any *two* vectors in (the same) vector space.

The displayed equation even shows the second fundamental form as a function of two variables — but nevertheless calls it a "quadratic form".

These concepts should not be confused. Especially not in a Wikipedia article. 2601:200:C082:2EA0:64D9:5FBE:7ECB:BC01 (talk) 03:07, 2 March 2023 (UTC)

Question on the second fundamental form for a hypersurface in Euclidean space
In the article it is said that in Euclidean space, the second fundamental form is given by


 * $$\mathrm{I\!I}(v,w) = -\langle d\nu(v),w\rangle\nu$$

where $$\nu$$ is the Gauss map, and $$d\nu$$ the pushforward (differential)|differential of $$\nu$$ regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space.

Probably, I am just not seeing something obviuos but for me this looks like a type mismatch. The map $$d\nu$$ is a vector valued $$1$$-form. Now, I plug in a vector $$v$$ to obtain a vector, which means that $$d\nu(v)$$ is a vector in Euclidean space. Now, $$w$$ is also a vector in Euclidean space, so the scalar product $$\langle d\nu(v),w\rangle$$ is a scalar. And now I multiply this scalar by a vector valued function $$\nu$$, so the final result $$-\langle d\nu(v),w\rangle\nu$$ is a vector. But if I understood the second fundamental form correctly, its values should be scalars. So, is there an error on the page or am I missing something? 2A00:1398:4:A09:4947:A324:CE18:4954 (talk) 09:25, 26 September 2023 (UTC)