Non-linear forms & tensor densities

[Rasalhague]

Tensor densities are normally defined in terms of coordinate transformations. Could they also be defined as functions of p tangent vectors and q cotangent vectors, just as tensors are defined, except relaxing the condition of linearity?

Can anyone suggest a good, basic introduction to non-linear forms such as the integrands called "elements" (line, area, volume). Are these a subset of tensor densities? David Bachman's A Geometric Approach to Differential Forms discusses these briefly in an appendix, but having read it, I still don't feel confident of what all the notation represents or what moves are allowed. For example, in his equation for a line element in orthonormal coordinates, does

$$\sqrt{\sum_{i}(dx^i)^2}$$

mean

$$\sqrt{\sum_{i} dx^i \otimes dx^i} \enspace ?$$

If so, does $\sqrt{dt \otimes dt} = dt$ mean $r \; \circ \; q$ where $r$ is the square root function, and $q$ the quadratic form derived from $dt \otimes dt$ by constraining this tensor to take the same input in each slot?

EDIT: In the last paragraph, the scope of the square root should be the whole expression, thus (dt (x) dt)^1/2. Unfortunately the line across the top didn't show up in the inline LaTeX.

 

[arkajad]

Tensors can be defined as sections of the associated bundles to the frame bundle of the manifold. They correspond to the natural representations of GL(n,R) on tensor products of R^n and its dual. If you add the factor det(A)^w or |det(A)|^w in these representations - you get tensor densities. Volume or surface elements can be understood this way.

 

[Rasalhague]

Suppose I have a function of p elements of TM at some point of M, and q elements of T*M at that point, with whatever properties of good behaviour are normally assumed for tensors, like a tensor in every way, in fact, except not linear because I've constrained two or more of its argument slots to take the same tangent vector or cotangent vector as input.

(1) Is this object necessarily a tensor density?

(2) Are there other kinds of tensor density that don't fit this description? For example, can the square root function be included in the definition of such a function and it still be a tensor density?

In other words, can tensor densities be defined like this in a coordinate independent way, i.e. without defining them in terms of how their representations change with a change of basis?

 

[arkajad]

Think of the simplest case: scalar density, for instance a volume element. It is not a function on tensors. It is a function on frames.

 

[arkajad]

Added:

You can write

$$g_{\mu\nu}=e^a_\mu e^b_\nu \eta_{ab}$$

or, using tensor product notation

$$g=e^a\otimes e^b \eta_{ab}$$

Then you can try to play with something like

$$\sqrt{|g|}=\sqrt{|e^a\otimes e^b \eta_{ab}|}$$

But this is just a notational game trying to hide the real nature of $$\sqrt{|g|}$$ which is a function on frames.

 

[Rasalhague]

Interesting. The volume element can be expressed as a triple wedge product of basis cotangent vectors with some coefficient. Cotangent vectors and wedge products of them are defined as functions of tangent vectors. So I guessed that the volume element must also be, in some sense, a function of vectors. In justifying the idea that covariant alternating tensor fields (differential forms) are an important kind of integral, Bachman begins with an integral which involves an area function, "a function which takes two vectors and returns a real number", p. 15. Bachman's tangent vectors here play a similar role to the tangent vectors in this article from Mathworld about the volume form. But a covariant tensor is supposed to take so many tangent vectors as its input, and return a scalar as its output. The volume element on the Mathworld page doesn't get contracted by the vectors at all, it ends up with exactly the same valence it started with, only expressed in a different basis field. Bachman's area element could be seen as a volume element contracted by one tangent vector, that being the "normal tangent vector" given by the cross product of the new basis tangent vectors for the submanifold. But he characterises it rather as a function of two vectors, these being the inputs to "norm of cross product" function--in which case it looks more like Mathworld's volume element or your idea of these things as being functions of a frame (coordinate basis field). Come to think of it, the tangent vectors that appear in his examples of integrals of covariant alternating tensors also seem not to contract them. Do your remarks about tensor densities apply equally to differential forms then? Are all of these objects to be regarded as functions of basis fields?

 

[Rasalhague]

The reason I guessed tensor densities might be like tensors except nonlinear is that Bachman begins,

A differential form is simply this: an integrand. [...] One word of caution here: not all integrands are differential forms. In fact, in most calculus classes we learn how to calculate arc length, which involves an integrand which is not a differential form. Differential forms are just very natural objects to integrate, and also the first that one should study. As we shall see, this is much like beginning the study of all functions by understanding linear functions. The naive student may at first object to this, since linear functions are a very restrictive class. On the other hand, eventually we learn that any differentiable function (a much more general class) can be locally approximated by a linear function. Hence, in some sense, the linear functions are the most important ones. In the same way, one can make the argument that differential forms are the most important integrands.

He then discusses surface area and arc length in an appendix called "Non-linear forms". Arc length, at least, has some relationship to a quadratic form. An area element is a kind of volume element (a 2-volume element). And a volume element is said to be a tensor density. This led me to wonder what the relationship was between the set of other integrands (besides alternating covariant tensor fields, "differential forms"), and these non-linear integrands called elements, and tensor densities.

 

[arkajad]

Do your remarks about tensor densities apply equally to differential forms then? Are all of these objects to be regarded as functions of basis fields?

Of course. All tensors and tensor densities can be regarded as equivariant functions on the frame bundle. Each of them is characterized by a different representation of the group GL(n,R).
For tensors (in particular for forms) this is just an equivalent description to the usual one via tensor products of tangent and cotangent bundles. But for tensor densities this is essentially the only way.

 

[arkajad]

In his classical book "Varietes differentiables, formes courants, formes harmoniques" de Rham is trying to avoid the concept of a density of a given weight, but in Chapter 2.5 he de facto introduces them when he defines hi "Odd differential forms" - they transform like ordinary forms, but there is an extra sign of the Jacobi determinant entering in the transformation law.
Treating all geometric objects as equivariant vector-valued functions on the frame bundle really makes things more clear.

 

[Rasalhague]

In the sense that a tensor or tensor density at each point takes tangent and cotangent basis vectors as input, and outputs its own scalar "components" in the basis appropriate to its valence?

EDIT: Oh, I didn't see your last post before I posted this. How is every tensor and tensor density vector-valued? Is this a vector in the general sense, or the particular sense of a tangent vector? Could you give an example?

 

[arkajad]

In the sense that a tensor or tensor density at each point takes tangent and cotangent basis vectors as input, and outputs its own scalar "components" in the basis appropriate to its valence?

Yes. Let $$\rho$$ be a representation of GL(n,R) on a vector space V (usually we take V to be a tensor product of R^n and its dual, or a subspace, for instance only symmetric etc.). Then you define tensor of type $$\rho$$ to be a function t on frames (at a given point of your manifold) that is equivariant of type $$\rho$$

i.e.

$$t(eA)=\rho(A^{-1})t(A),\, A\in GL(n,R)$$

Take $$V=R^n$$, take $$(\rho(A)v)^i=A^i_j v^j$$

and you get tangent vectors.

Take $$V=R^{n*}$$ and $$(\rho(A)v)_i=(A^{-1})_i^j v_j$$

and you get 1-forms (cotangent vectors).

Here $$(eA)_i=e_jA^j_i$$ is the right action of GL(n,R) on frames.

 

[jasomill]

Loomis and Sternberg cover the basic idea of densities in a decently geometric and concrete manner in Advanced Calculus, available for free on http://www.math.harvard.edu/people/SternbergShlomo.html" . Volume I of Spivak's Comprehensive Introduction to Differential Geometry covers them in a more abstract fashion, mostly in the exercises. Sternberg takes a rather nice "concrete approach to abstract differential geometry" in his Lectures on Differential Geometry, and certainly addresses integration of densities.

Focus on the ideas, not the notation — for a simpleminded take on "surface area integrals" written by a naive calculus student trying to work through these things, see attached. No claims of rigor or "technical correctness," and it certainly doesn't address G-structures or frame bundles, but it might help relate what you already know to what you're trying to learn. Or it might confuse you even more — who knows?

In the words of Hilbert,

"The art of doing mathematics consists in finding that special case which contains all the germs of generality."

For an glimpse at the foundations of the paradigm shift from "infinitely small parallelograms" to "bundles of frames," check out Élie Cartan's Riemannian Geometry in an Orthogonal Frame.

Finally, the cited de Rham book — chapter 1 is about the best self-contained treatment of the basic foundations of differentiable manifolds I've seen, and more generally a tour de force of high-powered mathematical exposition at its finest (he gets from the basic definitions through Whitney's embedding theorem in about 10 pages, without slowing down or missing a beat) — has been translated into English as Differentiable Manifolds: Forms, Currents, Harmonic Forms (Springer, 1984). Sadly, it appears to be out of print, but check your local university library.

Cheers,
Jason

 

[Rasalhague]

You've given something to think about, arkajad. Often the more recent stuff I've read emphasises a need to define things not in terms of bases or coordinates, but your description of tensors and tensor densities as functions of bases seems to relate more directltly to what Bachman is actually doing with what he calls differential forms and non-linear forms.

Thanks for the leads, jasomill. I'll check those out.

 

[Fredrik]

Lee's Introduction to smooth manifolds explains densities well. (Unfortunately one of the pages isn't available for preview).

 

[arkajad]

You've given something to think about, arkajad.

I thought I will give you even more to think about enlarging the scope to introduce also non-linear geometrical objects.

All takes place in a fixed point of the manifold and its tangent space at that point. We have a distinguished set of bases. On a generic manifold it will be the set of all linear bases. On a Riemannian manifold - orthonormal bases etc. So we have a group that acts on these bases. It is convenient to consider right action:

$$(eA)_i=e_j {A^j}_i.$$ The matrices A are in our group G. Be it GL(n,R), SO(1,3) or whatever. Then we take a space V on which this group acts from the left, so we have

$$\rho(A):v\mapsto \rho(A)v$$.

If V is a vector space this will be usually the standard linear representation of G, possibly with powers of the determinant (which would be irrelevant for SO(m,n) groups). But V can be also a manifold, for instance a homogeneous space G/H. For instance in the case of SO(1,3) it can be the 2-dimensional sphere.

The we define a geometric object of type $$\rho$$ to be an equivalence class of pairs (e,v), where the equivalence relation is defined by

(e,v) is equivalent to (e',v') if and only if there exists A in G such that

$$e'=eA,\, v'=\rho(A^{-1})v.$$

The equivalence class of (e,v) is written as e.v

Let $$\xi$$ be an equivalence class. Because it is assumed that the action of G on frames is transitive, for every e there is a unique v such that

$$\xi=e.v$$

We call this particular v the coordinate representation of $$\xi$$ with respect to the frame e. So we can write

$$\xi=e.v[e]=eA.\rho(A^{-1})v[e]$$

The we have

$$v[eA]=\rho(A^{-1})v[e]$$

All know geometrical objects. vectors, tensors, forms, densities, all of them fall into this description. But also non-linear objects can be studied, with values on a sphere or some other manifold. Covariant derivatives can be also calculated for these nonlinear objects (except that for nonlinear fields the covariant derivative of a V-valued field will be in the tangent space $$T_vV$$. For a linear space V we have that $$T_vV$$ can be identified with V).

This covers geometrical objects of the first order. There are also geometrical objects of second order (Levi-Civita connection), which are not tensorial, but this is another subject not relevant now.

All of that can be found in good books on differential geometry (like Sternberg), and it is indeed a useful perspective that clarifies lot of possible issues.