Contact geometry

In mathematics, contact geometry is a fascinating area that studies special structures on smooth shapes called smooth manifolds. It looks at something called a hyperplane distribution within the tangent bundle and focuses on a special rule known as 'complete non-integrability'. This means the distribution behaves in a very specific and interesting way.

Contact geometry is closely related to another branch of math called symplectic geometry, but it works mainly with shapes that have an odd number of dimensions, while symplectic geometry deals with even dimensions. Both of these areas are inspired by the ideas used in classical mechanics. In mechanics, scientists often look at something called phase space, which describes all the possible states a system can be in. Contact geometry, however, often looks at a special slice of this space that has one fewer dimension.

This field helps mathematicians and scientists understand complex systems and movements, making it an important tool in both pure math and practical physics.

Mathematical formulation

Contact geometry studies special structures on smooth surfaces in higher dimensions. It focuses on a particular way of arranging "hyperplanes" — flat spaces of one dimension less than the whole space — so that they never line up in a predictable way. This makes the geometry behave in interesting and useful ways.

A contact structure is like a rule that tells us how these hyperplanes are placed at each point in space. This rule makes sure the hyperplanes are always "non-integrable," meaning they don’t fit together to form larger flat surfaces. This property is what gives contact geometry its unique characteristics and applications.

Examples

The standard contact structure

The standard contact structure in R³ , with coordinates (x,y,z), is the one-form dz − y dx. The contact plane ξ at a point (x,y,z) is spanned by the vectors X₁ = ∂_y and X₂ = ∂_x + y ∂_z.

These planes appear to twist along the y-axis. This example generalizes to any R²ⁿ⁺¹ . It is standard, because Darboux's theorem states that any contact structure is locally the same as the standard one.

The standard contact structure on the sphere

Given any n, the standard contact form on the _(2n-1)_sphere S^2n-1 is obtained by restricting the Liouville 1-form λ = Σ i ( x_i d y_i − y_i d x_i ) on R²ⁿ to the unit sphere. Equivalently, it is obtained by the Liouville 1-form on Cⁿ .

The Reeb vector field is Σ j=1^n ( x_j ∂ y_j + y_j ∂ x_j ) = Σ j=1^n ( z_j ∂ z_j + _z¯j ∂ _z¯j ) , which generates the Hopf fibration.

Equivalently, consider the standard symplectic structure ω = Σ i d x_i ∧ d y_i on R²ⁿ . Each 1-dimensional subspace V is isotropic, and has a complementary coisotropic subspace V^ω that contains it. Projectivized to P(R²ⁿ), each point in P(R²ⁿ) has a complementary plane that contains the point. This distribution of planes is isomorphic to the standard contact structure on S^2n-1 .

One-jet

Given a manifold M of dimension n , the one-jet space J¹(M,R) is the space of germs of type M → R identified up to order-1 contact. Intuitively, each point in J¹(M,R) is a mapping from an infinitesimal neighborhood of M to R . Each member of the space can be identified by the three quantities x ∈ M , f(x) ∈ R , ∇f(x) ∈ T_x* M , thus J¹(M,R) is a manifold of dimension 2n+1 and can be identified with T*^* M × R . It has a natural contact form α = df − θ given by the tautological 1-form θ = Σ i=1^n y_i d x_i . The standard contact structure is the special case where M = Rⁿ .

Any first-differentiable function M → R then uniquely lifts to a Legendrian submanifold in J¹(M,R) , and conversely, any Legendrian submanifold is the lift of a first-differentiable function M → R . Its projection to M × R is the graph of the function. This also shows that J¹(M,R) embeds into the contact bundle of hyperplane elements C_n(M × R) , defined below.^: 311 

Contact bundle of hyperplane elements

Given a manifold M of dimension n + 1 , its n-th contact bundle C_nM is the bundle of its dimension-n contact elements. More abstractly, it is the projectivized cotangent bundle C_n(M) ≅ P(T*^* M) . Locally, expand M in coordinates as q⁰, … , qⁿ , then the contact bundle locally has coordinates ( q⁰, … , qⁿ, [ p₀, … , p_n_ ] ) , where p₀, … , p_n_ uses projective coordinates. Any n-submanifold of M uniquely lifts to an n-submanifold of C_nM . Conversely, an n-submanifold of C_n(M) is a lift of an n-submanifold of M iff it annihilates the 1-form Σ μ=0^n p_μ_ d q^μ . On the subset where p₀_ ≠ 0 , the condition becomes d q⁰ + Σ i=1^n p_i_ d qⁱ , which is the standard contact structure.

Similarly, the contact bundle of cooriented hyperplane elements C_n(M)⁺ ≅ S(T*^* M) is obtained by spherizing the cotangent bundle, i.e. quotienting only by R⁺ .

The contact structure on C_n(M) can also be described coordinate-free. Define π : C_n(M) → M to be the fiber projection that maps a hyperplane element to its base point. Then, for any ξ ∈ C_n(M) , a local tangent vector v ∈ T_ξC₁(M) is a simultaneous translation of the base point and a rotation of the hyperplane element. Then v is in the hyper-hyperplane at ξ iff π(v) is in the hyperplane element of ξ itself. In other words, the 2n-dimensional hyper-hyperplane at ξ is spanned by translation of the base point within ξ , as well as rotation of the hyperplane element while keeping its base point unchanged.^: 311 

Be careful with two meanings of hyperplanes here. A hyperplane element on M is an infinitesimal dimension-n hyperplane in M . These are the points of the contact manifold C_n(M) . The contact structure of C_n(M) consists of hyperplane elements in C_n(M) , which are infinitesimal dimension-2n hyperplanes in C_n(M) . The contact structure is not over M , which can have even dimensions, whereas C_n(M) necessarily has odd dimensions.

When M = R² , C₁M is the contact bundle of line elements in the plane, and is homeomorphic to the direct product of the plane with the projective 1-space R² × P(R¹) . The contact structure of C₁(M) looks like plane elements that rotate around their axis as they move along the "vertical" P(R¹) direction, completing a 180° when it finishes one cycle through P(R¹) . The standard contact structure in R³ can then be induced via a map R³ → R² × P(R¹) . Equivalently, the contact structure on C₁(M) can be constructed by gluing R³ at infinity. However, whereas the contact structure on R³ is coorientable, that on C₁(M) is not, since of P(R¹) is not orientable. It can be double-covered by C₁(M)⁺ ≅ R² × S¹ , which is coorientable.^: 8  A circle in the plane lifts to a helix in C₁(M)⁺ , but a double helix in C₁(M) .

Others

Until the 1950s, the only contact manifolds were the above ones, until Boothby and Wang in 1958 made a general construction via contactization.

The Sasakian manifolds are contact manifolds.

Brieskorn manifolds are defined by Σ ( a₀, … , a_n) = { ( z₀, … , z_n) ∈ Cⁿ⁺¹ ∣ z₀^a₀ + ⋯ + z_n^a_n = 0 } ∩ S²ⁿ⁺¹ where the a_j are natural numbers ≥ 2 and S²ⁿ⁺¹ is the unit sphere in Cⁿ⁺¹ . It has a contact structure defined by i/2 Σ j=0^n ( z_j d z¯_j − z¯_j d z_j ) = 0 .

Every connected compact orientable three-dimensional manifold admits a contact structure. This result generalizes to any compact almost-contact manifold.

Contact transformation

A contact transformation is a special kind of mapping between two contact manifolds that keeps their contact structure the same. This means that the way lines and surfaces touch each other remains unchanged after the transformation.

There are different types of contact transformations. One important type is called a strict contact transformation, which needs a specific choice of contact forms to be defined. Another type involves infinitesimal contact symmetries, which are linked to vector fields that generate small changes in the contact structure. These ideas help mathematicians study the properties of contact manifolds and their transformations.

Submanifolds

In contact geometry, we study special types of submanifolds within a contact manifold. These include contact submanifolds, which are submanifolds where a certain condition is met, and isotropic submanifolds, where the tangent space at each point lies within a specific distribution.

Another important type is the Legendrian submanifold. These are very common and follow a rule called an h-principle. This means that locally, any Legendrian submanifold can be described using simple functions. For example, in a contact 3-manifold, a Legendrian knot is a closed curve that follows these rules. Even though different Legendrian knots might look the same as smooth knots, their behavior can be quite rigid, meaning there are many different ways they can be embedded that are all smoothly connected but not all stay Legendrian during the process.

Vector fields

Liouville

In a symplectic manifold, a vector field is called Liouville if it satisfies a special condition related to the symplectic form. This condition helps identify the manifold with a standard one.

A Liouville form is a 1-form that, when differentiated, creates a symplectic form. The tautological 1-form is an example of this.

Reeb

Main article: Reeb vector field

Given a contact form on a manifold, it has a Reeb vector field, or characteristic vector field. This vector field is uniquely defined and shows that the paired hyperplane elements are preserved under Reeb vector flow.

The Reeb vector field is not part of the contact structure but rather of the contact dynamics. If a contact form arises as a constant-energy hypersurface inside a symplectic manifold, then the Reeb vector field is the restriction to the submanifold of the Hamiltonian vector field associated with the energy function.

The dynamics of the Reeb field can be used to study the structure of the contact manifold or even the underlying manifold using techniques of Floer homology such as symplectic field theory and, in three dimensions, embedded contact homology. The Reeb field is named after Georges Reeb.

Relation with symplectic geometry

There are many ways contact geometry and symplectic geometry connect, often inspired by physics. Since a symplectic form deals with even dimensions and a contact form with odd dimensions, any link must bridge these dimensions. This means a relationship usually exists between a contact manifold of dimension (2n - 1) or (2n + 1) and a symplectic manifold of dimension (2n).

One key idea is "contactification," where a symplectic manifold can be turned into a contact manifold by adding a extra dimension. Another method is the "Liouville transversal construction," which creates a contact manifold from a special kind of subspace in a symplectic manifold. These connections help mathematicians study both geometries together, revealing deep links between them.

Main article: Symplectization

Topology

The topology of contact 3-manifolds is well understood. For any oriented 3-manifold, there are infinitely many different contact structures. These can be created by performing surgery along a Legendrian link. Some of these structures are called overtwisted, while those that are not are called tight. The standard contact structure on a sphere is the only tight one possible.

The Weinstein conjecture asks whether, on a compact contact manifold, any Reeb flow always contains a cycle. This has been proven true in the case of 3-dimensional manifolds. The Gray stability theorem shows that contact structures on closed manifolds cannot be changed into non-equivalent structures through continuous deformations.

History

The ideas behind contact geometry go back a long way, with early hints in the work of ancient and classical mathematicians like Apollonius of Perga, Christiaan Huygens, Isaac Barrow, and Isaac Newton. Later, the theory of contact transformations was developed by Sophus Lie, who used it to study important math problems and understand how space changes.

The term "contact manifold" was first used in a paper published in 1958.

Applications

Contact geometry, like symplectic geometry, has many uses in physics and other areas. It helps us understand things like light and sound waves, movement in physics, and even how heat works in different materials.

Scientists have used contact geometry to solve hard math problems about shapes in space. For example, they have figured out special properties of certain shapes and even how knots behave in three dimensions. This shows how useful contact geometry is in both natural science and pure math.