Tensor #37. Tacks and Sheaves, Honeycombs텐서 #37. 압정, 다발, 벌집

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 38 of the playlist, Tacks and Sheaves, Honeycombs. In the previous post we drew a vector as an arrow and a dual vector as a stack. This post draws new pictures made by combining two arrows and two stacks.

An operation must be invariant

The dual space mapping $a \cdot b$ of the previous post had the same value when the basis was stretched. This is because the components of the stack grow and the components of the arrow shrink, and they cancel. This property is our guide for this post. An operation is meaningful only when it gives the same answer however we change coordinates, and whenever we build a new object we check how that object changes when the basis size changes.

Combining two arrows

Consider an operation that takes two arrows and gives something. If the value is a real number, it is an element of a $(0,2)$ tensor product space. But it does not have to be a real number, only invariant. In elementary physics we made the cross product of two arrows $a, c$ into an arrow $d$. The arrow $d$ is perpendicular to $a, c$ and its length is the area of the parallelogram they make. This needs an inner product for perpendicular to have a meaning. Without an inner product we cannot speak of perpendicularity. Moreover, if we scale the basis size up by $u$, each of the two sides has its components multiplied by $1/u$, so the component of the area is multiplied by $1/u^2$. The components of an arrow change by $1/u$, so this result does not change like an arrow.

So Weinreich makes a new object and calls it a thumbtack. A thumbtack is the parallelogram made by two arrows, taken together with a direction. The shape does not matter, and only the area and direction are fixed. Whether stretched long or pressed short, if the area is the same it is the same thumbtack. It is a small piece of area with a size, quite different from a stack, which extends planes infinitely.

In the language of this series, a thumbtack is an element of $\V \wedge \V$. In three dimensions this space is also three-dimensional, with basis $e_1 \wedge e_2, \; e_2 \wedge e_3, \; e_3 \wedge e_1$. These thumbtacks are the unit area pieces in the basis directions, including the parallelogram made by $e_1, e_2$, and a general thumbtack is a linear combination of the three. The components come from measuring the area in each basis direction. This is the same way that the arrow components of the previous post were lengths in basis directions and the stack components were densities in basis directions. When the basis is scaled up by $u$, the thumbtack components are multiplied by $1/u^2$, and they change by the rule of an element of $\V \wedge \V$, which differs from the rule for an arrow.

That $\V$, $\V^*$, and $\V \wedge \V$ are all three-dimensional is a coincidence. It is because $\binom{3}{2} = 3$ happens to match in three dimensions, and in other dimensions it differs.

Adding thumbtacks and scalar multiples

A thumbtack is an element of a vector space, so we can add them and multiply by constants. A scalar multiple scales the area by that constant. Addition is done geometrically. Turn two thumbtacks so that they share one side, and the parallelogram made by joining the remaining sides is the sum. As a formula, $(a \wedge b) + (c \wedge b) = (a + c) \wedge b$. (This formula is added for understanding and was not covered in the lecture. The lecture shows it only as a picture.) Two thumbtacks with opposite directions have their areas cancel when their sides are put together, and the sum is the zero thumbtack.

Q. If the shape of a thumbtack is not fixed, may we still draw it as a parallelogram?
A. Yes. Weinreich draws a parallelogram to show that it was made from two arrows. A thumbtack carries only area and direction, so any shape with the same area and direction is the same thumbtack. Comparing areas and adding them is done by fitting the shapes.

Orientation and the right-hand rule

To put an arrowhead on a thumbtack we use the right-hand rule. This is an arbitrary choice. Mathematics does not say which of the right and left hand is correct, so it is not canonical. What this choice amounts to is orientation. When the basis is placed in the order $e_1, e_2, e_3$, we declare the orientation positive if the component of the top-degree form $e^1 \wedge e^2 \wedge e^3$ is positive. In three dimensions the space of 3-forms is one-dimensional, so the component is either positive or negative. If positive the orientation is right-handed, and if negative it is left-handed. However many bases we pick, each falls into one of two classes, the positive class or the negative class.

Once an orientation is fixed, the right-hand rule lets us turn a circulation mark into an arrowhead. There are cases where an orientation cannot be given. The Möbius strip is one, and there we would have to use pictures with a circulation mark (a curlicue) in the place of an arrowhead. This lecture treats only manifolds and spaces with an orientation given and uses arrowheads in every picture.

Weinreich’s cross product is the wedge product

What Weinreich calls the cross product is our wedge product $a \wedge b$. He defines the result as a new object, but for us the wedge product already exists as an operation and its result is the new object. Our definition is more fundamental.

To receive the result as an arrow, as in the cross product of elementary physics, we need the Hodge dual. In three dimensions the dimensions of $\V$, $\V \wedge \V$, $\V \wedge \V \wedge \V$ are $3, 3, 1$. The Hodge dual sends a thumbtack to an arrow only when a metric and an orientation are given. So the cross product in textbooks is the wedge product followed by the Hodge dual. MTW does not name the thumbtack separately and draws two vectors $u \wedge v$ as a parallelogram. It is a picture that does the same job.

Q. It would be convenient to take the result of the cross product as an arrow, so why make a separate thumbtack?
A. That arrow is fixed only when an inner product exists, and when the basis size changes it changes differently from the arrow rule. A thumbtack is fixed without a metric, and its transformation rule also fits.

Two stacks and the honeycomb

This time we combine two stacks. A thumbtack was made by leaving two arrows in place and completing a parallelogram. We do the same with stacks. The addition of the previous post was finding one new set of planes from two sets of planes. This time we leave the two sets as they are and look at their whole intersecting structure. When two sets of planes overlap we get a structure of square columns (tubes) lined up, and MTW calls this a honeycomb.

This honeycomb also has a direction. Turn the circulation from the first stack to the second stack into an arrowhead by the right-hand rule, and the way the columns extend is fixed. This object is $\alpha \wedge \beta \in \V^* \wedge \V^*$, that is, a 2-form. In three dimensions this space is also three-dimensional and has addition and scalar multiplication.

The component of the honeycomb is the density of cells in the cross-section of the columns, that is, how many cells there are per unit area. If both stacks are dense the cells are small and the density is high, and if one is sparse the cells are large. And as the directions of the two stacks get closer, the density approaches 0. It is the same property as a thumbtack, where two arrows nearly on one line give a parallelogram of almost zero area even when the sides are long. A thumbtack was the area made by two arrows and a honeycomb is the density made by two stacks, so they are partners.

The sheaf

Weinreich does not use the honeycomb. He drops the outlines of the cells, takes only the lines where the two sets of planes meet, and puts an arrowhead on each line by the right-hand rule. A set of lines with arrowheads like this is called a sheaf, and he defines it as the result of combining two stacks. It has no relation to the word sheaf used in mathematics.

If we move the lines from the intersections to the centers of the cells, we get the same sheaf. It becomes a picture with one line threaded through each column. With the honeycomb we count cells and with the sheaf we count lines, and the count per unit area is the component. The number of thin lines passing through a piece of area does not change under continuous deformation of the space. A line outside the boundary of the piece cannot come inside by deformation. Drawing field lines of the electromagnetic field in physics is exactly the sheaf picture, and we could draw the same thing with a honeycomb.

The mapping between thumbtacks and honeycombs or sheaves

Now we put a thumbtack against a honeycomb or a sheaf. The honeycomb is an element of $\V^* \wedge \V^*$, and in finite dimensions this is the dual space of $\V \wedge \V$, where the thumbtack lives. So a dual space mapping is defined, and it gives a real number, the number of honeycomb cells passing through the thumbtack. When the basis size changes, the area component of the thumbtack shrinks and the density component of the honeycomb grows, so the cell count stays the same. Seen as a sheaf the value is the same, and it is the number of lines piercing the thumbtack. The density of the lines is sensitive only to size changes in the direction perpendicular to the lines.

Just as the mapping between an arrow and a stack is a single real number, here too the value is a single invariant real number. In components it reads as follows.

(This formula is for the wedge product defined as $\alpha \otimes \beta - \beta \otimes \alpha$, a factor $1/2$ may appear depending on the textbook, and it was not covered in the lecture.) In two dimensions the right side becomes $\det(\alpha, \beta) \cdot \det(a, b)$, the product of the cell density and the area.

The honeycomb and the sheaf are two pictures that read the same mapping differently. In three dimensions a volume form turns a 2-form into a vector, and what we get is a vector density of weight 1. The sheaf is precisely this vector density. In the same way the thumbtack corresponds to a dual vector of weight $-1$, that is, a covector capacity, and the two are isomorphic. Isomorphic means that adding two elements and then transferring them gives the same result as transferring each and then adding. MTW stays on the honeycomb side and Weinreich on the sheaf side because the two pictures are isomorphic, and for those used to field lines in physics the sheaf is the familiar one. (This explanation in terms of vector densities and covector capacities turns into components what the lecture only names, and the concrete component formula $B_{ij} = \varepsilon_{ijk} S^k$ was not covered in the lecture. Here $B$ is the 2-form and $S$ is the vector density.)

Q. Are the honeycomb and the sheaf the same object?
A. They are elements of two different but isomorphic spaces. The honeycomb is a 2-form and the sheaf is a vector density. The real number we get by putting them against a thumbtack is the same, so choosing a picture is a matter of taste.

In the toy below, make a thumbtack with the blue and green points. Compare the count of cells that the honeycomb underneath passes through it with $\rho \cdot A$, and check that the product stays the same when you change the basis size.

Summary

What we confirmed in this post is the following.

• If we take the cross product of two arrows as an arrow, we need an inner product and the transformation does not fit either. So Weinreich makes a new object, the thumbtack. A thumbtack is a piece of area with a direction, and in our language an element of $\V \wedge \V$.
• A thumbtack has addition and scalar multiplication, and once an orientation is fixed we put an arrowhead on it by the right-hand rule. The orientation is an arbitrary choice fixed by the sign of the top-degree form.
• The combination of two stacks is a 2-form, drawn as a honeycomb by MTW and as a sheaf by Weinreich.
• The mapping between a thumbtack and a honeycomb or sheaf is a dual space mapping and an invariant real number, the number of cells or lines passing through the thumbtack.

With arrows, stacks, thumbtacks, and honeycombs (sheaves), we have nearly all the objects on the vector side. The remaining combinations among these objects are taken up in the lectures that follow.