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June 20

First fundamental form and curvature

I wanted to ask you for help in better understanding two concepts that leave me a little perplexed regarding the first fundamental form:

1) The first fundamental form was defined to me as follows: given a parameterization ϕ(u,v) , the metric A is found with the dot products like ϕu⋅ϕv (through the 4 permutations of u,v), where the dot products are restricted to the tangent spaces induced by R3. It was then explained that quantities dependent on E,F,G (such as the Christoffel symbols, and therefore the curvature via Gauss's Theorema Egregium since it shows that they depend only on the Christoffel symbols) are intrinsic quantities, meaning they do not depend on how the surface is immersed in R3, but are intrinsic to the object itself.

My confusion revolves around this: if I define the first fundamental form in this way, I note that ϕ(u,v) is the map ϕ:U→R3, so what comes out of this map is precisely the figure I have as a surface in R3R3, that is, the shape it takes. In fact, the map gives me the coordinates (x,y,z). Now, ϕ and ϕv are the tangent vectors to that figure, so they indeed depend on the shape realized in R3. Then I define the matrix A through the dot products of ϕu and ϕv, so what is intrinsic here? I am using tangent vectors to a figure that has a shape given by ϕ:U→R3, so I would say it indeed depends on the immersion and how the figure is geometrically realized in it.

What seems to be suggested by the explanation is this: if I immerse the "abstract concept" of a sphere in R3, I have different realizable figures. I do not understand this concept given the considerations above: curvature depends only on E,F,G through the Christoffel symbols, but E,F,G depend on the dot products of tangent vectors to a shape of the surface, so on an immersion of it in R3.

2) The second question is this: if I change parameterization, I will have new ϕi',i∈u,v which differ from the initial ones, so I will find E',F',G'. However, for an object (let's take the usual sphere), the curvature is fixed and depends only on E,F,G right? Well, if I have changed parameterization and have E',F',G', why wouldn't the curvature change? They are different values. It seems to me that by changing parameterization, the first fundamental form changes and therefore the curvature should change as well (but it shouldn't obviously be so).

Could you help me with these two questions? Thank you. --151.36.108.141 (talk) 15:42, 20 June 2024 (UTC)[reply]

E,F,G are relative to a particular tangent space and the values will of course change if one changes the tangent space. One has to get rid of the tangent space dependency to get things like the curvature. The determinant gives a measure of the area of dudv and dividing the determinant of the second fundamental form by that of the first fundamental form gives something independent of dudv - and which in fact is the Gaussian curvature. NadVolum (talk) 15:09, 21 June 2024 (UTC)[reply]