In this short post I want to introduce an instance of certain objects that will be the subject of a few more posts. This particular object arises naturally in Affine Differential Geometry and turned out to have a relevant rôle in Harmonic Analysis too (in both Fourier restriction and in the theory of Radon transforms).
1. Affine Invariant measures
Affine Differential Geometry is the study of (differential-)geometric properties that are invariant with respect to . A very interesting object arising in Affine Geometry is the notion of an Affine Invariant Measure. Sticking to examples rather than theory (since the theory is still quite underdeveloped!), consider a hypersurface sufficiently smooth to have well-defined Gaussian curvature, which we denote by (a function on ). If we let denote the surface measure on (induced from the Lebesgue measure on the ambient space for example, or by taking directly , the restriction of the -dimensional Hausdorff measure to the hypersurface) then this crafty little object is called Affine Invariant Surface Measure and is given by
It was first introduced by Blaschke for (finding the reference seems impossible; it’s [B] in this paper, if you feel luckier) and by Leichtweiss for general . The reason this measure is so interesting is that it is (equi)affine invariant in the sense that if is an equi-affine transformation (thus with and so volume-preserving since ) then, using subscripts to distinguish the two surfaces, we have
for any measurable . We remark the following fact: that seemingly mysterious power in the definition of is the only exponent for which the resulting measure is (equi)affine-invariant.
Although this elementary fact is extremely well known, my lazy search could not come up with a decent reference for it (there is a slick proof when is a convex body but the fact is much more general than this), so in the rest of the post I will prove the fact in a gruesomely direct way.
2. Affine invariance of
I don’t know whether there is a clever proof of the invariance. There probably is one, but I will rather prove the fact doing some pedantic hard work.
Instead of proving just (1), we will consider more in general what happens to under an affine transformation , where we omit the translation (since its action is trivially invariant) and consider the more general case of instead of just . Our claim is that
Before we start proving things, we list all the objects we will need.
2.1. Objects and notations:
We will use subscripts to denote which surface every object is referring to.
We let denote any point on the hypersurface and we let denote the normal at point according to the chosen (local) orientation. The tangent space at is denoted by – it is the space of vectors orthogonal to . The shape operator is the linear operator (where the latter is identified with since they are parallel) defined by
where is the directional derivative in direction . The Gaussian curvature of the hypersurface at point is then defined to be
where the determinant makes sense because we can see as a linear map . However, it is a bit difficult to deal formally with this abstract notion of determinant – we’d rather work with the usual determinant on matrices. There is a simple way to do this using projections. We let denote the orthogonal projection onto the normal , that is
similarly, we let denote the orthogonal projection onto the tangent space . Observe that . If we define to be the extension of to the whole of given by
then the Gaussian curvature can be calculated as
where now the determinant refers to the standard determinant of matrices.
All the definitions above have their counterparts in the hypersurface , clearly. To compute we will need to compute separately how and behave under the transformation , which is what we will do next.
2.2. Change in the normal direction under
The first thing we need to figure out is what is. This is easy: first of all, by linearity of , the tangent space is equal to (as vector sub-spaces). Therefore any can be written as for some . Secondly, the normal is the only unit vector (up to a sign) that is orthogonal to all of and so
It will be convenient in the following to have worked out what the corresponding projection is, in terms of and . Using (3) we have for any
2.3. Change in the surface area under
The next step is to calculate how is related to . If we let denote an orthonormal basis of , then the density of with respect to is given by the ratio
where the vertical bars denote the determinant of the matrix whose columns are the vectors inbetween the bars. Notice that the denominator is equal to 1 and so it suffices to evaluate the numerator. Thanks to (3) and the properties of the determinant we can write
2.4. Change in the Gaussian curvature under
This is the most cumbersome part of the calculations. We have to evaluate the shape operator in terms of and . Taking we have that for some and so by the properties of the derivative operator
Using (3) and linearity we have therefore
which by Leibniz’s rule is equal to
The numerator of the first term in square brackets we can readily recognise: it is simply . As for the numerator of the second term in square brackets, we can see that
(in the last line we have used that ). Combining this with the above we have shown that
At this point we can recognise in the above identity the projection as given by (4), so that we have
We are nearly there. At this point we remember that we want to use the extensions and in order to take advantage of the properties of the standard determinant, so we need to insert them in the expression above somehow. The first thing to notice is that we can replace above with for free, since . The second thing to notice is that we can sneak in any multiple of for free as well, because
Indeed, this follows from the almost obvious fact that (omitting unnecessary notation)
Therefore we can replace in the expression above with for free, and as a consequence we have the beautiful identity
Since , it remains to evaluate the determinant of the RHS. This can be done in a general way that is extremely reminiscent of what we did to calculate how the surface measure transforms under .
Let , let be an arbitrary unit vector and let be the orthogonal projection onto and let be the orthogonal projection onto the space orthogonal to . What we want is to compute
in terms of and . If we let be an orthonormal basis of we have that
Using the properties of the determinant we have therefore
We will apply this to above, for which we have to take
Since we have using (3) again
where the last line is because acts like the identity on and therefore . We have thus shown that
2.5. Change in under – conclusion
and therefore (2) is proven.