Geometric imagination of differential forms - MathOverflow

Geometric imagination of differential forms

2010-06-03T17:34:31Z

In order to explain to non-experts what is a vectorfield, one usually describes an assignemnt of an arrow to each point of space, and this works quite well, also when moving to manifolds (where a generalized arrow will be a tangent vector).

My question is: What are similar objects that can help imagining differential forms?

Positive qualities for such object would be (for example) that it helps justifying easily change of coordinate formulas and formulas for pullbacks via functions, or that it "easily drawable", or that it helps understanding more complicated differential-form-based concepts (e.g. connections, cohomology groups, etc.).

Answer by Deane Yang for Geometric imagination of differential forms

2010-06-03T17:49:04Z

If you look at Misner-Thorn-Wheeler's book, Gravitation, you will see heroic attempts to draw pictures of differential forms. But I think this is misguided. Not everything can be drawn directly as a picture.

To me, a 1-form is a measuring instrument defining a unit of speed for a vector field. Without a $1$-form, there is no natural way to measure the length of a vector field or the speed of its integral curves. A 1-form is the simplest co-ordinate-free way to do so.

Higher exterior powers of vectors and forms have corresponding but more elaborate explanations.

Answer by coudy for Geometric imagination of differential forms

2010-06-03T18:09:58Z

One-forms are coordinates on the tangent space at a given point. Given a basis of a n-dimensional vector space, the dual basis is just the nth coordinates functions on the vector space in the given basis. Of course, if the basis of vector depends on the point, its dual basis also changes as we move the point on the manifold.

n-forms in n-dimensional space are a way to compute volume, or more generally the extensive quantities that appears in physics: mass, electric charge etc. The "value" of the n-form at a point is the infinitesimal mass located at the point.

Explaining forms of intermediate degree is a bit tricky. I think that some physical intuition can help again. Given a thin surface of non-homogeneous material in 3-dimensional ambient space, we want to describe how the mass is distributed on the surface. At each point of the surface, we want a 2-dimensional element of "volume". This infinitesimal mass depends on the point chosen on the surface. The distribution of 2-dimensional mass can be described by a 2-form. That form retains two informations: the way the tangent plane to the surface is oriented in space (which is completely determined by the normal vector to the surface), and the infinitesimal mass value.

Arguably, such an explanation suggests that a p-form can be "integrated" so as to give some p-dimensional manifold, which is completely wrong. Still it provides some intuition for p-forms, which are meant to be integrated on manifolds, and it can be a starting point for a more elaborate answer.

Answer by Mariano Suárez-Alvarez for Geometric imagination of differential forms

2010-06-03T18:13:14Z

$p$-forms are what you can integrate over submanifolds of dimension $p$. While this may sound way too operative to be called an intuition, it'll get you quite far.

Answer by David Carchedi for Geometric imagination of differential forms

2010-06-03T21:22:49Z

A one-form assigns to each vector tangent to a manifold a real number in a linear way. You may think of a vector tangent to a manifold as being determined by two points on the manifold that are "infinitesimally close", and hence view a 1-form as a function from such infinitesimal pairs of points to the real numbers. This analogy persists in higher dimensions; an n-form is a way of assigning n-points which are mutually infinitesimally close a real number (with the additional assumption that this assignment be antisymmetric). This is more than an analogy. The techniques of synthetic differential geometry allow for a rigorous definition of n-forms this way and it can be proven to line up with the classical definition. The details can be found here:

http://home.imf.au.dk/kock/van00.PDF

Anders Kock has many of his papers on his webpage including some which explain the concept of connections in this language. The advantage of the synthetic approach is that the definitions seem to line up with heuristic arguments often used to think about these objects. Here is the webpage:

http://home.imf.au.dk/kock/

Answer by babubba for Geometric imagination of differential forms

2010-06-03T21:37:17Z

I'm sure Mircea you've already thought of what I'm about to say, but if you cheat a little (viz. you assume you're manifold has a metric) you can always think of cotangent vectors as tangent vectors which act on others by scalar product.

I guess as one imagines the wedging of tangent vectors as (higher dimensional) parallelograms one can apply a similar trick.

p.s. Another idea might be to view cotangent vectors as their kernels, so hyperplanes in tangent space. This gives a picture of the projectivisation of the cotangent space. Maybe this trick can be extended as well to differential forms?

Answer by SB for Geometric imagination of differential forms

2010-06-04T03:57:07Z

90% of the time, all you need to know about k-forms is that they are something that turns a k-tuple of vector fields into a function on your manifold. (Aka they eat vector fields and give functions.)

Answer by Greg Kuperberg for Geometric imagination of differential forms

2010-06-04T04:50:55Z

The $k$-forms that are easiest to describe are those with $k \in {0,1,n-1,n}$. A 0-form on an $n$-manifold is a function. A 1-form on an $n$-manifold, if you imagine it in $n+1$ dimensions, is like an arrangement of shingles on a roof: At each point of the manifold, it defines a directional slope, which as other people have said, is the same as a dual vector on tangent vectors. An $n$-form is a density, i.e., an entity that you can integrate over the manifold. And an $(n-1)$-form is a flux (like, say, describing oil coming out of a well): At each point it has a null tangent direction, and it assigns a non-zero volume to each cross section.

Of course you can think of any $k$-form as a $k$-dimensional flux, and for general values of $k$ you might as well. But when $k$ is 1 or $n-1$, it is somewhat easier to visualize the condition that the form is closed. A 1-form is closed when the shingles locally mesh as the slope of a smooth roof, i.e., the form is locally integrable. An $(n-1)$-form is closed when the flux is locally conservative, which for instance is the case with fluid flow. In fact, theorem: A closed, non-zero $(n-1)$-form is equivalent to a 1-dimensional foliation with a transverse volume structure.

The reason that other values of $k$ are harder is that while you do get an entirely analogous algebraic integrability condition when the form is closed, you might not get the same kind of geometric integrability. A non-zero 1-form has an $(n-1)$-dimensional kernel at each point. (Although the visualization that I suggested is in $n+1$ dimensions, it is also true in $n$ dimensions that these tangent hyperplanes mesh when the 1-form is closed.) A non-zero $(n-1)$-form has a 1-dimensional kernel at each point. But a $k$-form for other values of $k$ doesn't usually have a kernel. (Okay, a maximum rank 2-form in odd dimensions also has a 1-dimensional kernel, and it is equivalent to a 1-foliation with a transverse symplectic structure.)

I have heard the statement that only 1-forms and 2-forms are any good. (Well, that's an overstatement, but they are more important than the others except for maybe $0$ and $n$.) In particular, symplectic forms show up a lot, so it is important to try to imagine them even though by definition they have no kernels. I think of a symplectic form as a calibration for a local complex structure. (Or an almost complex structure, which might be all that exists globally.) I.e., among the different tangent 2-planes of a symplectic $2n$-manifold, the ones that are complex lines have the greatest pairing with the symplectic form, while the ones that are real planes have vanishing pairing, and the pairing minimum is achieved by complex lines with the wrong orientation.

One more remark: The geometric picture of a foliation with a transverse volume structure holds for closed $k$-forms that are also non-zero simple forms (i.e., wedge products of linearly independent 1-forms). I think it's a theorem that any closed $k$-form is locally a sum of closed, simple $k$-forms. If that's correct, then that's also a way to visualize a closed $k$-form, as an algebraic superposition of volumed foliations. $k=1$ and $k=n-1$ are special cases in which every non-zero form is simple.

Answer by Jon Awbrey for Geometric imagination of differential forms

2010-06-04T15:50:11Z

I personally learned a lot from trying to formulate and visualize the analogue of differential geometry over $GF(2)$, also known as differential logic. There is an exposition here.

Answer by Suresh Venkat for Geometric imagination of differential forms

2010-07-07T01:08:22Z

Terry Tao has wonderful article in the PCM on differential forms. I frequently refer to it when I'm trying to get my head around some concept.

http://terrytao.wordpress.com/2007/12/25/pcm-article-differential-forms/