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I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second fundamental forms are could be good. Then, what do its coefficients E, F, G and e, f, g mean? Last, could you explain the intuitive/geometric meaning/interpretation of the Christoffel symbols?

I would like to concentrate on two-dimensional surfaces, but further explanation pertinent to or in the context of higher-dimensional manifolds could be additionally helpful to me.

kevin
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    The first fundamental form is an intrinsic quantity, the second is an extrinsic. In other words, the 1st fundamental form is something a creature embedded in the space can calculate and understand whereas the 2nd fundamental form requires you to add a dimension to this space to calculate more interesting things like curvature. The coefficients you are talking about come from the surface patch you are given (I am not sure if this is your question). – qqo Apr 17 '14 at 06:03
  • It is partially. But what do the coefficients have to do with actual geometric meaning or interpretation? Are they meaningful at all? For example, the first fundamental form is the distance on the surface, so what is E in terms of determining that distance? Is it how distance is measured along the first direction? (If so, how is F important? Would not E and G be enough to specify how they mix, intuitively?) – kevin Apr 17 '14 at 07:17
  • (Obviously, E and G are not enough and F is needed, but as far as intuition goes, I do not get it if that thought was correct. It seems like one only needs to know how the basis directions are measured in order to calculate the length of any hypotenuse in them. But my plane geometry may be influencing that (F=0)) – kevin Apr 17 '14 at 07:21
  • The coefficients depend on the choice of coordinate system. I am inclined to say that means they have no geometric meaning. – Zhen Lin Apr 17 '14 at 09:19

2 Answers2

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Intuitively, the first fundamental form tells you how to compute the distances along the paths within the surface (it is just a Riemannian metric of the surface thought as a standalone manifold, that is if we forget about the embedding/immersion). This explains why it is also called the intrinsic metric.

The second fundamental form describes how "curved" the embedding is, in other words, how the surface is located in the ambient space. It is a kind of derivative of the unit normal along the surface or, equivalently, the rate of change of the tangent planes, taken in various directions within the surface. Alternatively, it is called the shape tensor (it has a close relation to the shape, or Weingarten, operator), and is an extrinsic quantity in the sense that it depends on the embedding.

The Bonnet theorem (see a discussion here) ensures that (under certain conditions) these two fundamental form uniquely characterize the surface (locally), that is we can "integrate" them to a piece of surface in the space uniquely up to a rigid motion of the space.

The bottom line is that the Ist and IInd fundamental forms are as good as a complete set of local invariants of a surface, and thus they are extremely useful and important in differential geometry.

Remark 1. With regards to the coefficients, the comments have fully addressed this question: they are just components of these tensors in a coordinate patch.

Remark 2. The Christoffel symbols is a coordinate way to represent the invariant differentiation of vector (and all tensor) fields along the surface that arises from the given structures. In our case we have the usual (standard, Euclidean) metric in the ambient space and the Levi-Civita connection of this metric is just the usual (flat, Euclidean) derivative (just partial derivatives of the component in the standard coordinates). This (ambient) connection has its own Christoffel symbols but in our setting they all are zero, so it is customary not to mention them. Taking a vector field tangential to the surface we can try to differentiate it with this ambient derivative but for this to work we need to extend this vector field off the surface. The result of the differentiation will certainly depend on the extension but the tangential part of this result turns out to be independent of extensions when restricted to the surface. This way we obtain the covariant derivative (of tangential vector, tensor, ... fields) in the surface, and the Christoffel symbols that you may have met are the "components" of this covariant derivative (the Levi-Civita connection of the first fundamental form).

Community
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Yuri Vyatkin
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  • If the second fundamental form is said to be 'trivial', does that mean it vanishes or that it has some trivial form? – Tom Jan 07 '20 at 00:56
  • @Tom I am afraid that here is not an appropriate place to ask this question as it has nothing to do with the topic. Honestly, I have never seen this word usage of "trivial" applied to the second f.f. in serious literature, but it may have both meanings. It would be better if you asked a separate question and provided some context. – Yuri Vyatkin Jan 07 '20 at 01:36
  • The word 'trivial' is applied to the second fundamental form in this sense in a paper by Schoen and Yau which contributed to Yau being awarded the Fields Medal: I would say that counts as serious literature. – Tom Jan 07 '20 at 18:50
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    @Tom Sure, it does. I have not read that paper, sorry. It would be very kind of you if you provided a precise reference details for those who might me interested in our discussion. It would be even better, if you created a separate question in this website regarding this issue, and kindly left a link, so that everyone could continue talking about this in the right place. Also, I apologize for an incorrect usage of the word "serious". – Yuri Vyatkin Jan 08 '20 at 02:47
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Let $\mathbf{r}$ be a vector in the containing space, e.g., $(x,y,z)$ in three dimensions. Let $\mathbf{w}$ be coordinates in the embedded manifold, e.g., $(u,v)$ in a two-dimensional manifold. Assume that components of $\mathbf{r}$ can be written as functions of $\mathbf{w}$, and vice versa. Then the metric tensor is defined as

$g_{ij} = \sum_k \frac{dr^k}{dw^i} \frac{dr^k}{dw^j}$

where letter superscripts are indices, not powers. $g$ is clearly symmetric.

The square of the arc length is $ds^2 = \sum_{ij} g_{ij} dw^i dw^j$.

In two dimensions, $ds^2 = g_{11} du^2 + g_{12} du dv + g_{21} dv du + g_{22} dv^2$, or
$E du^2 + 2 F du dv + G dv^2$. This is the first fundamental form for a surface.

The components of the shape tensor are projections of second partial derivatives onto the unit normal. In two dimensions, a unit normal is the cross product of the tangent vectors, which are derivatives of $\mathbf{r}$ with respect to $u$ and $v$. The shape tensor is given by

$b_{ij} = \sum_k n^k \frac{\partial^2 r^k}{\partial w^i \partial w^j}$,

which is clearly symmetric. The distance from the surface at r+dr to the tangent plane at r is given by

$2D = \sum_{ij} b_{ij} dw^i dw^j$.

In two dimensions this is $2D = b_{11} du^2 + b_{12} du dv + b_{21} dv du + b_{22} dv^2,$ or $2D = e du^2 + 2 f du dv + g dv^2$. This is the second fundamental form for a surface.

A good reference is sections 32-36 of Vector and Tensor Analysis by Harry Lass, McGraw-Hill (1950).

nonreligious
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Van Snyder
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