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How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the first fundamental form, the arc length of a curve $\alpha:U\to S$ is said to be intrinsic because it can be computed with knowledge of only these coefficients as $$\int_0^t\sqrt{E(u^\prime)^2+2Fu^\prime v^\prime +G(v^\prime)^2}.$$But an ant living on the surface could not compute this value since $E,F,G$ are computed using points described by $3$-coordinates. The $3^{\text{rd}}$ coordinate does not exist to the ant.

If one lived in $\mathbb{R}^2$ (or a surface which they think is $\mathbb{R}^2$), and did not know of the existence of $\mathbb{R}^3$, they would have no way to compute this value. Would they?

I understand intrinsic meaning invariant under isometries, but I don't see how an intrinsic property can be computed or can exist without reference to some bigger space. To compute the Gaussian curvature at a point, you explicitly use the fact that the point is described by $3$ coordinates.

JonHerman
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    Actually, you can define a coordinate system on $S$ in terms of diffeomorphisms between open subsets of $S$ and $\Bbb{R}$ that "glue" on the intersection. You can then compute the arc length using these coordinates, which don't depend on any embedding of $S$ in $\Bbb{R}^n$. In simpler terms, though, you can take "intrinsic" to mean that the arc length doesn't depend on the choice of embedding of $S$ in $\Bbb{R}^3$, of coordinates on $\Bbb{R}^3$, or of parametrisation of $S$. – A.P. Jun 03 '15 at 21:44
  • My professor told us that intrinsic means that the quality only depends on the first fundamental form. – Memeozuki Jun 03 '15 at 22:19

2 Answers2

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An intrinsic property is one that can be defined only in terms of the first fundamental form.

Taking our surface element to be $$f:U \to {\mathbb{R}^3}$$ We can parameterize f in terms of just two arbitrary coordinates (Even though the surface element is embedded in 3-space you don't need 3 coordinates to describe it) $$f(u,v) = \left( {x(u,v),y(u,v),z(u,v)} \right)$$ We can than use this parametrization to define the first fundamental form

$${g_{ij}} = \left( {\begin{array}{*{20}{c}} E&F \\ F&G \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\left\langle {\frac{{\partial f}}{{\partial u}},\frac{{\partial f}}{{\partial u}}} \right\rangle }&{\left\langle {\frac{{\partial f}}{{\partial u}},\frac{{\partial f}}{{\partial v}}} \right\rangle } \\ {\left\langle {\frac{{\partial f}}{{\partial v}},\frac{{\partial f}}{{\partial u}}} \right\rangle }&{\left\langle {\frac{{\partial f}}{{\partial v}},\frac{{\partial f}}{{\partial v}}} \right\rangle } \end{array}} \right)$$

The arc length is than defined by just the first fundamental form and is therefore an intrinsic property $$d{s^2} = Ed{u^2} + 2Fdudv + Gd{v^2}$$

$$L = \int\limits_a^b {ds} = \int\limits_a^b {\sqrt {Ed{u^2} + 2Fdudv + Gd{v^2}} } $$

Conceptual Example:

If you want to think about it terms of some ant that exists only in 2-space: This ant could travel the distance of some arc on a surface element and measure this distance without any knowledge of the ambient space, only knowledge of a 2-coordinate system it defines on the surface element.

Mathematical Example:

The unit 2-sphere can be parametrized in terms of only 2 coordinates, the angles phi and theta. $$f(\varphi ,\theta ) = \left( {\cos \varphi \cos \theta ,\sin \varphi \cos \theta ,\sin \theta } \right)$$

We can than define the first fundamental form as $${g_{ij}} = \left( {\begin{array}{*{20}{c}} {{{\cos }^2}\theta }&0 \\ 0&1 \end{array}} \right)$$

And therefore the arclength as $$L = \int\limits_a^b {ds} = \int\limits_a^b {\sqrt {{{\cos }^2}\theta d{\varphi ^2} + d{\theta ^2}} } $$

Definition of arc length independent of any specific ambient space:

To define the length of some arc on a hyper-surface element independent of the ambient space in which the element is embedded you can use Riemannian Geometry.

Take the metric tensor: $${g_{ij}} = {e_i} \bullet {e_j}$$

And the Line Element/Riemannian Metric: $$d{s^2} = {g_{ij}}d{x^i} \otimes d{x^j} = g$$

We can than define Arc Length in a way that is the same for all spaces in which the hyper-surface element could be embedded: $$L = \int\limits_a^b {\sqrt {g} } $$

  • But in order to compute this length you used $\mathbb{R}^3$. Your computations depend on the function $f:U\to\mathbb{R}^3$. I've read many times that intrinsic properties are ones that can be computed independent of some bigger space. How do you compute the length of a curve on a surface $S$ without talking about $\mathbb{R}^3$. You can take the infimum of the distance function, but the distance function is dependent on values in $\mathbb{R}^3$. – JonHerman Jun 03 '15 at 23:37
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The components $E$, $F$, and $G$ of the first fundamental form on a smooth surface $S$ are indeed defined using an embedding of $S$ into a Euclidean space, but you can imagine a situation (such as a "physical theory" in the surface) in which three functions $E$, $F$, and $G$ (or $g_{11}$, $g_{12}$, and $g_{22}$) are associated to each coordinate patch on an abstract surface. For instance, on $\mathbf{R}^{2}$ you might have $$ E = G = \frac{4}{(1 + x^{2} + y^{2})^{2}},\quad F = 0. $$ In this setting, arc length makes sense, even though no embedding is specified.

Andrew D. Hwang
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