1 Foundations

Symplectic geometry studies smooth manifolds equipped with extra structure that encodes an oriented notion of area on each tangent plane. Unlike Riemannian geometry, it does not measure length or angle. Instead, it is built around a differential 2-form that is everywhere nondegenerate and closed. This setting is naturally even-dimensional and underlies the modern geometric formulation of classical mechanics.

1.1 Symplectic manifolds

A symplectic manifold is a smooth manifold of even dimension together with a symplectic form. The manifold may be compact or noncompact, connected or disconnected, but the defining structure must be smooth and globally defined. In practice, symplectic manifolds provide the ambient spaces in which one studies Hamiltonian motion, geometric constraints, and rigidity phenomena.

1.2 Symplectic forms

A symplectic form is a differential 2-form that assigns an oriented area to pairs of tangent vectors. It is skew-symmetric and varies smoothly from point to point. Because it is nondegenerate, it pairs every nonzero tangent vector with another vector producing a nonzero value for some choice, which ensures that the form contains genuine geometric information rather than collapsing directions.

1.3 Nondegeneracy and closedness

The two defining conditions of a symplectic form are nondegeneracy and closedness. Nondegeneracy means that the form does not vanish on any tangent direction in a trivial way, so the geometry has no null directions. Closedness means that its exterior derivative is zero, a condition that gives the structure strong local regularity and connects it to conservation laws and potential functions in mechanics.

1.4 Examples of symplectic structures

The standard example is the cotangent bundle of a smooth manifold, which carries a canonical symplectic form. Another basic model is even-dimensional Euclidean space with coordinates grouped into position and momentum variables. Complex manifolds may also admit symplectic forms, and many surfaces of dimension two are symplectic because any nowhere-vanishing area form on an oriented surface is symplectic.

2 Basic theorems

The foundational theorems of symplectic geometry show that, despite the global richness of the subject, the local form of a symplectic manifold is surprisingly uniform. These results also explain why global symplectic questions tend to focus on embeddings, neighborhoods, and deformation behavior rather than local coordinate classification.

2.1 Darboux’s theorem

Darboux’s theorem states that every symplectic manifold looks locally like the standard symplectic vector space. In suitable coordinates, the symplectic form has a canonical normal form with no local curvature-type invariants. This is one of the central features distinguishing symplectic geometry from Riemannian geometry, where local invariants appear immediately.

2.2 Moser’s stability theorem

Moser’s stability theorem describes when a smooth family of symplectic forms can be related by a family of diffeomorphisms. Under appropriate hypotheses, forms that vary through a fixed cohomology class are equivalent by an isotopy. The result is widely used to compare symplectic structures on compact manifolds and to prove deformation invariance statements.

2.3 Weinstein neighborhood theorem

The Weinstein neighborhood theorem gives a standard local model for a symplectic neighborhood of a submanifold with appropriate properties, especially Lagrangian submanifolds. It says that such a neighborhood is determined, up to symplectomorphism, by relatively simple normal data. This theorem is a key tool for analyzing how submanifolds sit inside a symplectic manifold.

2.4 Symplectic embedding results

Symplectic embedding problems ask when one symplectic manifold can be placed into another by a symplectomorphism onto its image. Unlike volume-preserving embeddings, symplectic embeddings are constrained by subtle rigidity phenomena. Such results often involve sharp inequalities and reveal that symplectic geometry remembers more than dimension and volume alone.

3 Hamiltonian mechanics

Hamiltonian mechanics is the classical mechanics framework most naturally expressed in symplectic terms. States of a system are represented by points in phase space, and time evolution is generated by a function called the Hamiltonian. The symplectic form organizes the equations of motion and gives a precise meaning to conserved quantities and canonical changes of variables.

3.1 Phase space formulation

Phase space combines position and momentum variables into a single geometric arena. A physical system is then described by functions on this space rather than by positions alone. Symplectic geometry supplies the natural structure on phase space, allowing trajectories to be viewed as geometric flows determined by energy.

3.2 Hamiltonian vector fields

Given a smooth Hamiltonian function, one can associate a Hamiltonian vector field through the symplectic form. This vector field generates the time evolution of the system and is defined so that the symplectic form converts the differential of the Hamiltonian into a direction of motion. The resulting flow preserves the symplectic structure.

3.3 Poisson brackets

The Poisson bracket is an algebraic operation on smooth functions that measures how observables change under Hamiltonian flow. It encodes the interaction between pairs of functions and satisfies identities reflecting the underlying geometry. In mechanics, it provides a compact way to write evolution equations and conservation relations.

3.4 Canonical transformations

Canonical transformations are changes of coordinates on phase space that preserve the symplectic form. They send one Hamiltonian description of a system to another without altering the essential mechanics. Such transformations are important in simplifying equations, identifying integrable systems, and constructing new coordinate systems adapted to conserved quantities.

3.5 Conservation laws and integrals of motion

A conserved quantity is a function that remains constant along a Hamiltonian trajectory. In symplectic terms, conservation often arises from symmetries and the structure of the Poisson bracket. Integrals of motion help reduce the effective complexity of a system and are central in the study of stability, integrability, and long-term behavior.

4 Submanifolds and local geometry

Symplectic manifolds contain several special classes of submanifolds that reflect how the symplectic form interacts with lower-dimensional geometry. These objects help describe constraints, symmetry reduction, and local models for neighborhoods. Their study is essential for understanding both the internal structure of symplectic spaces and the dynamics that occur on them.

4.1 Lagrangian submanifolds

A Lagrangian submanifold is a submanifold on which the symplectic form vanishes and whose dimension is exactly half that of the ambient manifold. Such submanifolds are among the most important objects in symplectic geometry. They arise naturally in mechanics, mirror symmetry, and the study of generating functions.

4.2 Isotropic and coisotropic submanifolds

Isotropic submanifolds are those on which the symplectic form restricts to zero, though they need not be maximal in dimension. Coisotropic submanifolds satisfy a complementary condition involving the symplectic orthogonal distribution. These classes generalize Lagrangian submanifolds and play roles in reduction procedures and constraint systems.

4.3 Symplectic reduction

Symplectic reduction is a method for constructing a lower-dimensional symplectic manifold from one with symmetry. By imposing constraints and quotienting by the symmetry action, one often obtains a new space that retains a symplectic structure. The procedure is fundamental in mechanics, geometric invariant theory, and the study of moduli spaces.

The Weinstein conjecture concerns the existence of periodic orbits on certain contact-type hypersurfaces and is closely tied to symplectic and contact methods. Related problems investigate characteristic foliations, Reeb dynamics, and the behavior of closed trajectories. These questions connect local geometric properties with global dynamical consequences.

5 Symplectic topology

Symplectic topology studies global properties of symplectic manifolds that remain under symplectomorphism. Its central theme is rigidity: many symplectic phenomena cannot be changed by smooth deformations in the way one might expect from ordinary topology. The field developed powerful techniques from analysis, topology, and dynamical systems to detect this rigidity.

5.1 Rigidity and flexibility

Symplectic geometry combines flexible local behavior with surprisingly rigid global constraints. Some structures can be deformed widely, while others are tightly restricted by invariants and embedding obstructions. Understanding which features are rigid and which are flexible is one of the main organizing questions of the field.

5.2 Symplectic capacities

A symplectic capacity is a numerical invariant that measures the size of a symplectic manifold in a way compatible with symplectic embeddings. Capacities are designed to detect obstructions invisible to ordinary volume. They are especially useful in proving that some regions cannot be symplectically embedded into smaller-looking ones.

5.3 Gromov non-squeezing theorem

The Gromov non-squeezing theorem is a landmark result showing that a symplectic ball cannot be “squeezed” into a cylinder of smaller radius by a symplectic embedding, even if their volumes would allow it. This theorem revealed that symplectic geometry has a strongly nontrivial notion of size. It was one of the first major demonstrations of symplectic rigidity.

5.4 Pseudoholomorphic curves

Pseudoholomorphic curves are maps from Riemann surfaces into almost complex manifolds that satisfy a nonlinear elliptic equation. Introduced by Gromov, they have become indispensable in symplectic topology. They provide analytic tools for defining invariants, proving compactness statements, and detecting geometric constraints.

5.5 Floer homology

Floer homology is an infinite-dimensional homology theory built from solutions of differential equations related to symplectic action functionals. It captures subtle information about intersections, periodic orbits, and the topology of symplectic manifolds. The theory has numerous variants and has become a major bridge between geometry, topology, and dynamics.

6 Symplectic invariants

Symplectic invariants are quantities or algebraic structures preserved under symplectomorphisms. They help distinguish manifolds that may be similar from a smooth or topological perspective but differ in symplectic behavior. Many of these invariants arise from counting geometric objects or from the analysis of Hamiltonian dynamics.

6.1 Gromov-Witten invariants

Gromov-Witten invariants count pseudoholomorphic curves in a symplectic manifold, subject to specified constraints. They provide powerful enumerative information and connect symplectic geometry to algebraic geometry. These invariants are central in modern intersection theory and play a major role in quantum cohomology.

6.2 Quantum cohomology

Quantum cohomology is a deformation of ordinary cohomology that incorporates counts of holomorphic or pseudoholomorphic curves. It enriches the classical cup product by adding symplectic curve-counting data. The resulting algebra often reflects deep geometric properties and links topology with enumerative geometry.

6.3 Hofer geometry

Hofer geometry studies the group of Hamiltonian diffeomorphisms using a natural metric defined from Hamiltonian functions. This metric reveals geometric structure on the group of symmetries generated by Hamiltonian flows. It has become an important tool for understanding energy, displacement, and the size of transformations.

6.4 Spectral invariants

Spectral invariants are numerical values extracted from Floer-theoretic constructions. They record subtle information about Hamiltonian diffeomorphisms and action spectra. These invariants are used in rigidity results, metric estimates, and the study of Hamiltonian dynamics on symplectic manifolds.

7 Relations to other fields

Symplectic geometry sits at a crossroads of several mathematical disciplines. Its methods overlap with analysis, topology, algebraic geometry, and dynamical systems, while also serving as a language for classical and modern physics. These connections have made it one of the most influential branches of geometry.

7.1 Differential geometry

Symplectic geometry is a major subfield of differential geometry, sharing techniques involving manifolds, tensor fields, and differential forms. At the same time, it differs from other geometric frameworks because its basic structure is not metric but skew-symmetric. This leads to different local normal forms and different types of global invariants.

7.2 Algebraic geometry

Many complex algebraic varieties carry natural symplectic structures compatible with their complex geometry. Conversely, techniques from symplectic topology often illuminate problems in algebraic geometry through curve counting and moduli spaces. The relationship is especially strong in areas such as mirror symmetry and the study of enumerative invariants.

7.3 Topology

Topology enters symplectic geometry through questions of classification, embeddings, and the behavior of submanifolds. Although symplectic structures are smooth objects, their existence and uniqueness are constrained by topological data such as cohomology classes. Topological methods are also essential in reduction, cobordism, and Floer-theoretic constructions.

7.4 Dynamical systems

Because Hamiltonian flows are dynamical systems on symplectic manifolds, symplectic geometry provides a natural framework for studying trajectories, stability, and recurrence. Periodic orbits, invariant sets, and action principles are all central topics. The field supplies geometric tools that sharpen the analysis of long-term behavior.

7.5 Mathematical physics

Symplectic geometry is deeply connected with classical and quantum physics. In classical mechanics, it gives the natural language for phase space and Hamilton’s equations. In more advanced settings, it supports the geometric formulation of field theories, quantization ideas, and structures appearing in modern theoretical physics.

8 Advanced topics

Advanced symplectic geometry extends the basic theory into richer interactions with contact geometry, categorical methods, and moduli problems. These topics often involve sophisticated analytic and algebraic machinery. They also connect symplectic geometry to cutting-edge research in topology, representation theory, and mathematical physics.

8.1 Contact geometry

Contact geometry is the odd-dimensional counterpart of symplectic geometry. It studies maximally nonintegrable hyperplane distributions and often appears as the boundary theory of symplectic manifolds. Many problems in symplectic topology have contact analogues, and the two subjects are tightly linked through dynamics and reduction.

8.2 Mirror symmetry

Mirror symmetry is a duality phenomenon relating symplectic geometry to complex algebraic geometry. On the symplectic side, it often involves counts of Lagrangian submanifolds and pseudoholomorphic curves. The theory has led to new predictions, new invariants, and deep correspondences between apparently distant geometric worlds.

8.3 Moduli spaces

Moduli spaces classify geometric objects up to an equivalence relation and often carry natural symplectic structures. Examples include spaces of flat connections, stable maps, and certain spaces arising from gauge theory. Symplectic methods help analyze their geometry, singularities, and deformation theory.

8.4 Fukaya categories

Fukaya categories are algebraic structures built from Lagrangian submanifolds and Floer theory. They encode intersection data, holomorphic curve counts, and composition rules in a categorical framework. These categories are central in homological mirror symmetry and have become a major organizing principle in modern symplectic geometry.

8.5 Symplectic field theory

Symplectic field theory is a broad framework that extends Floer-theoretic ideas to symplectic and contact manifolds. It uses moduli spaces of pseudoholomorphic curves to define rich algebraic structures and invariants. The theory aims to unify many previously separate constructions in symplectic topology and low-dimensional dynamics.