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The only reasonable way to interpret "$ds$" as a functional on tangent vectors has to be that it takes a tangent vector and spits out its length, but this is not linear. So $ds$ is not a 1-form. It still seems like a nice sort of object to think about integrating. Does $ds$ fit into a larger class of gadgets generalizing differential forms? Or it there some compelling reason that I shouldn't care about $ds$?

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What sort of properties of differential forms would you like this generalisation to have? –  Marco Golla Jun 13 '12 at 20:06
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I see a vote to close, and I find this very strange: this is a perfectly good question. (I vote to keep open) –  André Henriques Jun 13 '12 at 20:12
    
How do you even consider $ds$ as a function of the tangent bundle? The title suggests that you mean the expression $ds$ in an integral such as $\int \dot\gamma(s)ds$, right? This, however, is not "the same" $ds$ that people sometimes use to describe the local expression of a Riemannian metric, as in $ds^2 = dx^2 + dy^2$. Of course, if you mean the latter, then you indeed just have a family of euclidean norms, as the definition of Riemannian metric suggests. –  Malte Jun 13 '12 at 20:15
    
Contrary to Malte's comment above, the $ds$ in a scalar line integral is the same as the $ds$ in the expression of a Riemannian metric. As Álvarez-Paiva indicates in the currently accepted answer below, this object is an even density of rank $1$; on the one hand, such a thing may be multiplied by a scalar field to produce another even density of rank $1$, and such a thing may be integrated along any (unoriented!) $1$-dimensional submanifold (aka curve); on the other hand, such a thing may be multiplied (symmetrically!) by itself to produce (sometimes) a symmetric bilinear form. –  Toby Bartels Mar 3 '13 at 8:10
    
And as long as it is understood that we are using symmetrised multiplication (not the antisymmetrised multiplication that is the wedge product of differential forms), then the equation $(ds)^2 = (dx)^2 + (dy)^2$ is literally correct (for the Euclidean metric on the $(x,y)$-plane, which is literally $(dx)^2 + (dy)^2$). And even the formula $ds = \sqrt{(dx)^2 + (dy)^2}$ is literally correct and defines $ds$ as the unique positive semidefinite (which in this case is actually positive definite) square root of the Euclidean metric $(dx)^2 + (dy)^2$. –  Toby Bartels Mar 3 '13 at 8:15

4 Answers 4

up vote 32 down vote accepted

It is not a 1-form, it is a 1-density: a function that is continuous and homogeneous of degree 1 on the tangent space of the manifold. It also happens to be convex and positive in the complement of the zero section (actually, its restriction to each tangent space is a Euclidean norm). If the norm is not Euclidean, you have the arc-length element of a Finsler metric. The convexity is basically necessary and sufficient for the lower semi-continuity of the length functional (Busemann-Mayer theorem).

See my answer to this question for more on densities.

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Ok, I am sold . –  Steven Gubkin Jun 13 '12 at 20:23
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And integrals with respect to $ds$ make sense on any rectifiable curve, even those that are not differentiable. –  Gerald Edgar Jun 13 '12 at 20:24
    
Integrals with respect to act length are usually signed things, in that they take into account the orientation of the curve, no? That would suggest the $ds$ is a form then. –  Mariano Suárez-Alvarez Jun 13 '12 at 21:00
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@Mariano: No, I would say that they are not signed things. This is clearly visible in the notation $\sqrt{dx^2+dy^2}$. Of course, you can make them signed things if you want: given an orientation on a 1-dimensional manifold, you can turn a density into a one-form and vice versa. –  André Henriques Jun 13 '12 at 21:24
    
Gerald Edgar's comment is correct but may not be as big a generalisation as one might think, since every rectifiable curve may be parametrised by arclength, and this parametrisation is continuously differentiable almost everywhere, and the classical definition of the line integral of a continuos function using a Riemann integral and the derivative of the parametrisation therefore already applies (and gives the correct answer). –  Toby Bartels Mar 1 '13 at 19:52

It is an example of an absolute differential form, as defined by Toby Bartels here: http://ncatlab.org/nlab/show/absolute+differential+form.

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Interesting. I will have to look into this closer - I know anything by Toby Bartels is good. –  Steven Gubkin Jun 14 '12 at 2:25
    
These absolute forms are the same thing as densities (rediscovered). As far as I know, this concept dates back to papers by L.C. Young in variational calculus and varifolds (he called them generalized vaieties) and, independently, to Gelfand and Gindikin in integral geometry. –  alvarezpaiva Jun 14 '12 at 6:22
    
Actually, Bartles's definition is incomplete: at each point $p$ and for each $k$-tuples of linearly independent tangent vectors $v_1$, $\dots$, $v_k$, if $A: T_p M \rightarrow T_p M$ is an invertible linear map, then one needs either $\omega_p(Av_1,\dots, Av_k) = det(A)\omega_p(v_1,\dots,v_k)$ (odd densities that need to be integrated on oriented submanifolds) or $\omega_p(Av_1,\dots, Av_k) = |det(A)|\omega_p(v_1,\dots,v_k)$ (even densities). If you don't have this condition, the integral will depend on the parametrization of the submanifold. –  alvarezpaiva Jun 14 '12 at 6:33
    
Hmm, ok. Better update the nLab then. Thanks! –  David Roberts Jun 14 '12 at 6:57
    
I'm delighted to learn that my ‘absolute differential forms’ have been seen before! I'm also not surprised that my definition is too general; I was never really satisfied with it. I only wish that David had reported this conversation in the relevant thread on the nForum: nforum.mathforge.org/discussion/3744/…. (But that's not David's job either.) –  Toby Bartels Mar 1 '13 at 21:50

It is the volume element on the one manifold. It is a 1-form.

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In what sense can you say that $ds$ = $\sqrt(dx^2+dy^2)$? This is very common in high school calculus classes, and matches my intuition well, but it doesn't seem to make any sense from a differential forms perspective. Maybe this is more what I wanted to ask about. –  Steven Gubkin Jun 13 '12 at 19:58
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The volume element of the projective plane with its (let's say) standard Riemannian metric is a form ? –  alvarezpaiva Jun 14 '12 at 6:40
    
It should be a 1-pseudoform, just as the volume element of the projective plane must be a 2-pseudoform. Only with an orientation (not available for the projective plane) can one identify forms and pseudoforms. But this is still not a good answer, since we want to know what it is on the ambient space, not just what it is on the curve. –  Toby Bartels Mar 23 '13 at 5:33

If you have a curve, also known as a 1-manifold, inside a Riemannian manifold, the Riemannian metric on the manifold restricts to a 1-dimensional Riemannian metric on the 1-manifold. The square root of this metric is a density (see alvarezpaiva's answer) that can indeed be integrated along the 1-manifold.

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More generally, given a regular curve, i.e. an immersion $\gamma: [a,b] \rightarrow (M,g)$, one has the pullback metric $\gamma^\ast g$. The volume of $([a,b],\gamma^\ast g)$ is the length of the curve $\gamma$, which is $\int g(\dot\gamma(s),\dot\gamma(s)) ds. However, $ds$ here denotes the volume element of $[a,b]$ with the standard euclidean metric from the real line. –  Malte Jun 13 '12 at 20:26
    
This is also not a good answer, since we want to know what $ds$ is on the ambient space, not just what it is on the curve. –  Toby Bartels Mar 23 '13 at 5:34
    
Toby, you have a reasonable point. It is a 1-density on the ambient space that, when restricted to any curve and integrated, gives the length of the curve. Thanks. –  Deane Yang Mar 23 '13 at 13:59

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