Isabel’s post at God Plays Dice asking for a geometric entity realizing the dot product got me thinking again about a question that my friend Ron asked me several years ago: “Why do geometers always focus on antisymmetric tensors and ignore the symmetric ones?”
It certainly seems that antisymmetric objects have more grit to them — they correspond to things like fields of infinitesimal k-planes, or symplectic structures, or fluxes. Things you can imagine getting your hands on. Symmetric objects, on the other hand, are things like Riemannian metrics and the dot product, or the shape operator. They seem to measure things, which is far more subtle than being things. I’m always happier with a 2-form over a quadratic form: I can see the former.
There is a lot to say about this subject, and exploring it could take us to some fun places. As a first step in that direction, I’d like to share a proof that every manifold admits a metric which may be new to you.
One of the most-used tools in the differential geometry kit is the epic-sounding partition of unity. These let us glue things defined on a small-scale into things defined over your whole manifold, making many problems in geometry solvable by the schema “1.) do a thing locally, then 2.) glue it up using partitions of unity into a global thing”. Incidentally, the absence of holomorphic partitions of unity is what makes local/global problems on complex manifolds far less trivial than on smooth real manifolds. This is entirely responsible for the differences between the theory of smooth manifolds and the theory of complex manifolds.
The first serious application of partitions of unity is usually to prove that every manifold admits a nonvanishing symmetric 2-form — in other words, a Riemannian metric. The proof is easy once you can build and run the (nontrivial!) partition of unity machine: split your manifold up into chart domains, pull back the dot product on to get local metrics, glue them up into a global metric. There are details which I will happily ignore, because I want to talk about a different proof.
The previous proof is excellent for getting used to working with partitions of unity. However, it gives some false intuition. We should be surprised that every manifold admits a metric. Here’s why: a Riemannian metric is a nonvanishing symmetric 2-form on . By analogy, consider an ANTIsymmetric 2-form on . The kernel of this 2-form then defines a nonvanishing -plane field on . The world-famous hairy ball theorem tells us that many manifolds fail to admit nonvanishing vectorfields. Why would we expect them to admit nonvanishing -plane fields?
In fact, if is a nonorientable 2-manifold then it must not admit any nonvanishing 2-form fields, since a nonvanishing 2-form field would give a consistent orientation. So many manifolds will fail to admit nonvanishing antisymmetric 2-form fields — why would be expect them to admit nonvanishing symmetric 2-form fields?
Here is a fundamentally different proof which hopefully sheds a little more light on the subject, if you know your bundles. Consider the space of frames on . A point in is a pair where is a point of and is an ordered basis of the tangent space to at . This is a principal -bundle over . Picking a metric on is the same thing as saying which of these frames should be considered orthonormal. The different choices of “orthonormal” we could make at a point are parametrized by . So to pick a set of frames to call orthonormal over all of , we need to find a section of the bundle of orbits .
Now, most bundles just don’t have global sections. However, the Gram-Schmidt process gives us a retraction of onto the subgroup , so the quotient is homotopy-equivalent to a point. This means that there are no nontrivial bundles with fiber isomorphic to . We can always find a global section.
But then we are finished! A section of is the same as choosing an -orbit of frames at each point, which is the same a metric. The fact that retracts onto means that must be a trivial bundle. It therefore admits a global section, corresponding to a global Riemannian metric.
This phenomenon (symmetric data retracting away, leaving only topologically interesting antisymmetric data) appears in other places, and might be lurking behind our difficulties in realizing symmetric objects geometrically. More on these “other places” some “other time”.