What is so geometric about symplectic geometry?

Question

Symplectic geometry is often motivated by the Hamilton's equation which in turn are a reformulation of Newton's third law. But the subject itself is of independent mathematical interest. What I don't understand is why symplectic geometry is, in fact, geometry. Compared with Riemannian geometry where angles, lengths and geodesics can be defined and which can be used to provide models for classical geometry, the notion of a smooth manifold equipped with some two-form doesn't really seem geometric to me at all.

In fact I would argue that the aspect of providing a model for classical geometries is the defining feature of Riemannian geometry making it "geometry".

The only "geometric interpretion" of symplectic geometry I always see is "symplectic geometry is the geometry of phase spaces" with no mention what the geometric content of phase spaces actually is. Notions like a two-form or a Liouville form seem to me much more algebraic than geometric in nature.

Is there a more conceptual way of understanding symplectic structures by motivating them through geometric properties? Or perhaps even some axioms of some geometric space for which symplectic geometry provides a concrete framework to construct a model for?

A remark: One could answer this by defining geometry as the study of some space with an associated symmetry group (in the Riemannian case isometries and in the symplectic case symplectomorphisms), but this only moves the question to why one would care about symplectomorphisms from a geometric viewpoint. The case of isometries again is rather clear….

Answer 1

I think a mainstream answer would be that symplectic geometry has two (seemingly opposiate, but actually related) aspects: rigidity and flexibility, it is the rigidity aspect that makes symplectic geometry a kind of geometry.

The study of the rigidity of symplectic manifolds dates back to Gromov's groundbreaking work on $J$-holomorphic curves, where $J$ is an almost complex structure on the symplectic manifold $(M,\omega)$ tamed by $\omega$. Using this theory Gromov deduced a lot of interesting facts such as the non-squeezing theorem and the non-existence of simply-connected closed Lagrangian submanifolds in $\mathbb{C}^n$. So unlike topology, the appearance of a non-degenerate 2-form does impose restrictions when studying problems related to embedding, immersion, homotopy or isotopy.

The algebraic nature of symplectic geometry also originates from the existence and abundance of $\omega$-tame almost complex structures, since they have led to constructions of various algebraic structures associated to symplectic/contact manifolds like the quantum cohomology (there is an $E_2$-algebra structure on the chain level), Fukaya category (which is an $A_\infty$-category), linearized contact homology (there is an $L_\infty$-structure on the chain level)... In fact, the algebraic nature of symplectic geometry is usually the reason why many of the rigidity results should hold. For example, a lot of restrictions of Lagrangian embedding can be deduced from the classification of the objects in the Fukaya category, and this method is extremely effective, say, when the quantum cohomology is semisimple as a ring.

On the other hand, since Gromov's theory deals only with $J$-holomorphic curves with finite energy, so you can think of them as sort of more flexible analogues of algebraic curves. This shows that when studying its rigidity aspects, symplectic geometry behaves in some sense more like algebraic geometry. For example, it is a result due to Kollar, Tian, Starr, Ruan, etc. that uniruledness and rational connectedness of smooth projective varieties are invariant under symplectic deformations. Many invariants in algebraic geometry or singularity theory, say the log Kodaira dimension of a quasi-projective variety, or the minimal discrepancy of an isolated singularity can also in some sense be interpreted as symplectic/contact invariants.

Finally, I'd like to mention that when looking at symplectic manifolds from a more flexible perspective, symplectic geometry behaves more like differential topology. This is the reason why symplectic geometry is sometimes referred to as symplectic topology. Parallel to the theory of $J$-holomorphic curves, which treats symplectic manifolds as generalizations of algebraic varieties, when taking a handlebody decomposition perspective, one would naturally expect to relate the geometry of an open symplectic manifold with $H^k(M;\mathbb{Z})=0$ for $k>\frac{1}{2}\dim(M)$ to that of a Stein manifold, that's why Gromov and Eliashberg's h-principle plays a pivotal role there.

Answer 2

I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.

1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes. Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.

Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.

2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but I don't think it is much more geometric than

Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.

The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.

3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".

To see the "geometry" in symplectic geometry, you also have to look at its impressive array of applications to other geometries that may look more geometric to you at first, and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.

Answer 3

As pointed out in the comments, the term geometry can have many meanings. Given a symplectic manifold $(M,\omega)$, I think almost (you never know!) all of us will agree that a Riemannian metric $g$ on $M$ would qualify as a geometric structure. While the symplectic form $\omega$ on its own does not provide a metric, the notion of an $\omega$-compatible or $\omega$-tame almost-complex structure on $M$ serves as a link between the symplectic structure and interesting Riemannian structures. See, for example, Almost Complex Structures: 'Tame' versus 'Compatible'.
Perhaps one can say that the symplectic structure of $M$ — just like its topology — imposes restrictions on (reasonable/interesting/compatible) Riemannian metrics available on $M$, and can in this sense be regarded as encoding certain rough geometric features shared by all (reasonable/interesting/compatible) geometries available on $M$. Whether this makes the symplectic structure itself a geometric structure now depends on the roughness level of one's personal definition of geometry.

Answer 4

My take on this is fairly simple-minded.

A metric is a non-degenerate symmetric bilinear form on a vector space. This yields a notion of distance, angle and volume. Dually, we can define a 'cometric' as a non-degenerate cosymmetric - aka antisymmetric - bilinear form. This yields an area and volume form. Area and volume are eminently geometric notions. What it is showing is that area can be defined independently of any notion of angle and distance.

Of course the analogue of a 'cometric' over smooth manifold is not quite a symplectic form. We require it to be a closed form. But this is basically an integrability condition.

Of course 'cometric' is not the traditional name. I've used it to bring out the implicit duality here.

Further, the metric can be used to 'raise' and 'lower' tangent and cotangent fields, and we also have the metric Hodge star, gradient and divergence.

Dually, we can use the cometric also to 'raise' and 'lower' tangent and cotangent fields, and we also have a cometric Hodge star, gradient and divergence.