One of my summer projects was to try to learn symplectic geometry. In this, my first installment of notes, I discuss some introductory notions; hopefully it’s not too rambling. In the continuation, I’ll prove Darboux’s theorem, a fundamental result which says that locally, all symplectic spaces are isomorphic (something which sharply distinguishes symplectic geometry from Riemannian geometry, where there are many local invariants, such as curvature).
EDIT: Here, I take the point of view that the reader is somewhat familiar with Riemannian geometry, and try to build intuition for the structures in symplectic geometry via an analogy with the Riemannian case. This was helpful for me, to some extent, in order to even have a chance of “breaking into” the field, so to speak. However, there are many reasons why this is possibly a misleading vantage point, so do not believe that it’s the whole story. It may be helpful for some. Please see the comments for additional (undoubtedly better, I am an extreme novice) points of view. Some of these would also be quite suitable for an introduction.
In both symplectic and Riemannian geometry, the main object of study is a smooth manifold equipped with a bilinear form on each tangent space, in such a way that the forms vary smoothly as we move between tangent spaces. In the (possibly more familiar) Riemannian case, this form is a symmetric, non-degenerate, positive definite form, turning each tangent space into a normed vector space. In symplectic geometry, we instead require a skew-symmetric bilinear form on each tangent space, again varying smoothly. We still require that at each point in our manifold , should be non-degenerate, so that if =0 for all , then must be 0. Finally, note that because is a skew-symmetric 2-form, it is a differential 2-form on , and we require that as a 2-form, is closed, i.e., . I’ll introduce examples as we go.
Just like in Riemannian geometry, the fact that the symplectic form is non-degenerate establishes a natural bijection between the tangent space and cotangent space at every point. I’ll write this as . The image of is the functional which sends to . In the Riemannian geometry case, we know that, at a given point, we can just choose an orthonormal basis, and then we have a particularly nice “dual basis:” a basis vector is dual to the unique functional whose value on that vector is 1, and whose value on the other basis vectors is 0. This is a simple way to get a handle on the isomorphism from in the Riemannian case. In terms of this special basis, the symmetric form has been reduced to the standard dot product.
There is an equivalent construction in the symplectic case. The “standard” symplectic form on has a basis of the form , with pairing with to give 1 and pairing with the rest of the basis vectors (including itself) to give 0 (the form is determined by this data, along with the requirement that it be bilinear and skew-symmetric). There is a theorem, just as easy as Gram-Schmidt, that says that every symplectic form looks like the above in an appropriate basis (a corollary is that if a vector space can be equipped with a non-degenerate skew-symmetric form, then it is necessarily even-dimensional). Therefore after a change of basis, the bijection between tangent and cotangent space sends to the covector (linear functional) which is 1 on and 0 on the other basis vectors. As a side note, if we make into a complex vector space by letting , then the standard form can be written in terms of the standard dot product as , indeed, we have (I hope I have my signs ok). Thus the standard complex, Riemannian, and symplectic structures on all determine each other in this way, and make it into a Kähler manifold. More generally, complex or almost-complex structures compatible with symplectic forms are very important in the subject.
Before I move on to symplectic manifolds, note that just on the level of bilinear forms on vector spaces, there are many features of the geometry of skew-symmetric forms which are quite different than that of symmetric forms (to which we are probably more accustomed), and these all have important implications in the non-linear (= general symplectic geometry) case. For example, for , define . Then the non-degeneracy of the form insures that, just like for a normed vector space, the dimensions of and add up to . However they are not necessarily disjoint! Indeed, for example, with the standard form, . Such a subspace is called Lagrangian, meaning the form restricted to it is identically 0, and it is of maximal dimension () with this property.
I want to give some examples before I go further, but unfortunately, I haven’t really found any amazingly clear and enlightening ones. The simplest and most obvious is itself, with the standard form, which we have already discussed. Already in this case there are interesting questions to be asked. As another, note that if is a smooth orientable surface, then every 2-form is closed, as there are no non-zero 3-forms, and any volume form will be non-degenerate. Therefore all surfaces are symplectic manifolds, and any one-dimensional subspace will be Lagrangian. So until we either introduce some more interesting questions, or go to higher dimensions, we are a little stuck (I could also mention here that every cotangent bundle is a symplectic manifold in a natural way, but I’ll probably talk about this next time). The second thing I want to mention, because it bothered me at first, is the question of motivation for symplectic geometry. Certainly, the formalism arose from physics, and simplified many natural physical models. However, I like to think of it as just another “kind” of geometry on a smooth manifold, another fairly simple structure we can put on it—an evil twin of Riemannian geometry, for which we can ask all the same questions, and see what comes out. This point of view then justifies itself as we find quite elegant behavior, as well as applications to other fields (which I probably won’t discuss this time around).
I have to make a quick disclaimer right now. Throughout, whenever I talk about integrating a vector field to a flow, I’m going to assume implicitly (without stating!) that we’re on a compact manifold , so that the flow is well defined on all of . This is very sloppy: might as well call the rest of these notes “Stuff about compact symplectic manifolds.” Except that Darboux’s theorem is true in general, which I’ll indicate.
OK, on to geometry. To every smooth real-valued function on a smooth manifold , we can associated the differential , which is the one-form whose value at a vector field at a point is the value of the directional derivative at . The nice thing about having the dual map at each point is that from each differential 1-form we can produce a vector field, and this applies to , giving a vector field which depends on . Now when we do this construction with a Riemannian metric, we get (by definition) the gradient vector field, and it has the nice property that it is normal to the level sets of (indeed, it points towards the direction of most increase of , which naturally is perpendicular to the level sets). To see this, one just applies the derivation to , and unwinding the definitions one gets simply the smooth function , the norm squared of for each point . However, doing this with gives the directional derivative as ! Indeed, is tangent to the level sets of (so it is just rotated by in some direction, or, alternatively, hit by the multiplication by map, in an appropriate basis and some compatible complex structure). So the flow of any point under stays within a particular level set, and so the level sets of are invariant under the flow (the standard example is to take the sphere with its standard volume form as the symplectic form, and take the function to be the height function. Then the level sets are circles of constant height, and the flow of can be seen to be rotation around the vertical axis). This is perhaps the moment to mention that is called a Hamiltonian vector field (rather than gradient vector field as in Riemannian geometry), and is called the Hamiltonian (function) (of . Indeed, the fact that the level sets of are invariant under the flow is a hint at the physics origins (as is the name Hamiltonian) of the whole subject. Originally, one would take a particular Hamiltonian to be the energy on a particular phase space (which is itself defined as the cotangent bundle to a state space, giving it a natural symplectic structure, as I alluded to previously). Then one declares that the allowable evolution of the universe is to follow the flow of . That level sets are preserved under the flow gives conservation of energy. This formalism is called Hamiltonian mechanics.
To be continued! I’ll say a little more about Hamiltonian flows, and then prove Darboux’s theorem.