41

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….

LSpice
  • 11,423
Jannik Pitt
  • 1,162
  • 11
  • 23
  • 13
    Klein's Erlangen Program essentially equates geometry with invariant theory of some subgroup (the principal group) of the general linear group (the comprehensive group). Here the principal group would be the symplectic group. I think the question is related to extending Klein's dictionary from invariant theory to differential invariant theory. – Abdelmalek Abdesselam May 31 '21 at 18:48
  • 7
    It's not my area of research, but from the outside something like Gromov's non-squeezing theorem seems like a very geometric result. – RBega2 May 31 '21 at 18:50
  • 9
    A symplectic form is a closed non-degenerate 2-form. You can interpret this as a way of measuring 2-dimensional area for infinitesimal surface pieces. One is led to symplectomorphisms, as you say, which turn out to be more restrictive then asking for "volume preserving" maps (e.g. non-squeezing theorem). So there is some geometric content. – Ryan May 31 '21 at 18:53
  • 11
    Is projective geometry geometric enough? There are no angles, no distances, no lengths not even areas and volumes like in symplectic geometry. Perhaps you need to look at applications of symplectic and contact geometry to Riemannian geometry, optics, convex geometry, integral geometry, etc. so that you see the geometry come out. Just the basic definitions and the elegance of the Hamiltonian formalism won't give you any insight. – alvarezpaiva May 31 '21 at 19:39
  • To me geometry is the study of the properties of some kind of space, any set endowed with a sufficiently interesting structure deserving to be called that way. – Sylvain JULIEN May 31 '21 at 19:42
  • 1
    @SylvainJULIEN What is a "space"? Is then e.g. field theory also geometry? – Jannik Pitt May 31 '21 at 19:58
  • 17
    I think it's fair to say mathematicians use the word "geometry" generously. Traditionally there hasn't been a hard-and-fast rule for saying when something isn't geometric. You could extend your line of questioning further. Much of Riemannian geometry aligns with the study of PDEs. Does that mean PDEs should be viewed as the central to geometry? Or should we say much of Riemannian Geometry isn't geometric? Ultimately there aren't any strict rules applied to subject-area titles. I think we have a thread here somewhere, complaining about "K-theory" as a title, probably many others. – Ryan Budney May 31 '21 at 20:13
  • One could also wonder what a number is. Are quaternions genuine numbers? Their usefulness in physics gives them a "real" legitimity, even though Pascal said "les nombres imitent l'espace, qui est de nature si différente"... – Sylvain JULIEN May 31 '21 at 21:15
  • I do feel that true geometry (or at least true differential geometry) should have some notion of curvature (or at least flatness should be a non-trivial condition). In this sense, symplectic geometry does seem closer in spirit to topology. – RBega2 May 31 '21 at 21:31
  • 2
    Hamilton's equations and classical mechanics can seem very analytical. Symplectic geometry gives a very geometric way of thinking about them. For example, instead of "Hamilton's principal function" satisfying the Hamilton-Jacobi equation, you have a Lagrangian submanifold flowing under the Hamiltonian flow. Instead of a trajectory of the Hamiltonian system with given initial and terminal conditions, you have an intersection point between two Lagrangian submanifolds. – Jonny Evans May 31 '21 at 21:38
  • Kahler manifolds have symplectic forms, and the symplectic geometry captures a lot of deformation-invariant geometry, like the Gromov-Witten invariants. For example, the existence of 27 lines on a smooth cubic surface is pretty uncontroversially a geometry theorem, and you still have it in symplectic geometry. – Jonny Evans May 31 '21 at 21:39
  • Another example: if you have a Lagrangian torus fibration, the action coordinates give you an integral affine structure on the base, which is extremely rigid and geometric. Skew symmetric forms are not what we're used to and you have to work a bit to understand the geometry they give you. – Jonny Evans May 31 '21 at 21:41
  • 4
    A Thurston-inspired definition of "geometry" would come down to a measurement of what part of the brain you need to think about the subject. A subject is geometry in this sense if and only if you have to intensively use your visual cortex to think about the subject. I imagine there would be some variability from person to person and particular issues one might be engaged in thinking about, but I suppose this could be used as a starting point to make the word "geometry" more rigid. – Ryan Budney Jun 01 '21 at 02:15
  • 1
    @Ryan Budney this is fascinating do you have a reference that makes this definiton? I could imagine interdisciplinary work on this in psychology or neuroscience, that would be really interesting – Jojo Jun 01 '21 at 09:11
  • @joe: He goes into these sorts of ideas in a few places. But perhaps the place where he situates our modes of thinking about mathematics the most concretely is his "On proof and progress in mathematics" article. https://arxiv.org/abs/math/9404236 The "definition" I provide is an extrapolation from this -- I am not suggesting Thurston made this definition explicitly. But he certainly could have. – Ryan Budney Jun 01 '21 at 16:04
  • 4
    @JannikPitt The geometric aspects of the study of fields (and other commutative algebras, rings, schemes, etc.) is more commonly known as "algebraic geometry". – Terry Tao Jun 01 '21 at 16:56
  • 7
    Personally, I view the subject of geometry as lying on a spectrum of rigidity, with topology at the loosest end of the spectrum and homogeneous spaces (such as Euclidean space) at the most rigid end. Riemannian geometry is roughly at the midpoint, Kahler and algebraic geometry is nearer to the rigid end, and symplectic geometry is nearer to the looser end. But they all share similar themes, for instance topics such as symmetries, invariants, localisation, embedding, or dependence on dimension are common to all. – Terry Tao Jun 01 '21 at 17:09
  • 1
    A key example is the cotangent bundle, which is of course basic to study in differential geometry. I do that example along with Kahler manifolds give substantial geometric "bonafides". While this is at the opposite end of what the poster was asking about - axioms - I for one am satisfied by flowing from examples rather than axioms. – Dev Sinha Jun 01 '21 at 18:58
  • What is geometric about von Neumann's continuous geometry, or the geometry of noncommutative spaces? – Hollis Williams Jun 01 '21 at 21:26
  • This earlier question is closely related: https://mathoverflow.net/questions/283868/why-is-there-no-symplectic-version-of-spectral-geometry/283929#283929 – Dan Fox Jun 02 '21 at 09:34
  • @alvarezpaiva: in projective differential geometry there is an underlying connection on a principal bundle with finite-dimensional structural group. This is not so for symplectic manifolds without further structure. In this sense symplectic manifolds are akin to differentiable manifolds and the study of their global properties is akin to differential topology, so could be called "symplectic topology". Whether it is worthwhile to make clean terminological distinctions between geometry and topology is another matter (where would combinatorics fit?). – Dan Fox Jun 02 '21 at 09:40
  • 1
    @DanFox, it is nowhere fixed that geometry must be local. For most of its history it wasn't and even today local results in differential geometry are not usually the stuff of top tier journals. The fact that there are no local invariants in symplectic geometry does not mean it is not geometric in the same way that the non-locality (or infinitesimal character) of theorems in projective geometry does not mean it is not geometric. – alvarezpaiva Jun 02 '21 at 09:54
  • @alvarezpaiva: I would put it this way - the debate about how to use "geometry" may or not be a worthwhile enterprise - I'm inclined to be cautious about assigning the word any tight definition - but then the use of "geometry" in "symplectic geometry" and "riemannian geometry" is very different in some senses. That's fine, the use of "geometry" in "algebraic geometry" is probably closer to the symplectic sense than the riemannian sense. – Dan Fox Jun 02 '21 at 10:29
  • @alvarezpaiva: one confounding issue, relevant to the symplectic world, is where does volume fit? In the two-dimensional world area is part of geometry, whatever one means by geometry, In higher dimensions this is less clear. Perhaps one should speak also of "volumetric geometry". I think (maybe wrongly) of the "symplectic topology" terminology as due to Arnold. Note that Arnold/Khesin called their book "topological methods in hydrodynamics" not "geometric methods in hydrodynamics" and only write there of "geometry" when working with a metric on the diffeomorphism group. – Dan Fox Jun 02 '21 at 10:39
  • A huge portion of Riemannian geometry is just symplectic (the study of the Hamiltonian system described by its geodesic flow). Even the Levi-Civita connection is as easy to describe symplectically than it is to describe through the usual trick. See, for instance https://mathoverflow.net/questions/127319/intuition-for-levi-civita-connection-via-hamiltonian-flows/127394#127394 or https://mathoverflow.net/questions/256435/when-is-a-flow-geodesic-and-how-to-construct-the-connection-from-it/256484#256484 – alvarezpaiva Jun 02 '21 at 12:29
  • I think that what I object to is the "tight definition" of symplectic geometry. The study of the linear symplectic group, linear symplectic spaces, the Grassmannian of Lagrangian planes, the Maslov index and its variants, Hamiltonian systems, moment maps, etc., all that is symplectic geometry. – alvarezpaiva Jun 02 '21 at 12:43

4 Answers4

18

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.

YHBKJ
  • 3,157
11

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.

alvarezpaiva
  • 13,238
7

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.

LSpice
  • 11,423
B K
  • 1,890
3

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.

Mozibur Ullah
  • 2,222
  • 14
  • 21