Contact geometry

What is Contact Geometry?

In mathematics, contact geometry is a special area that studies shapes and their properties. It looks at smooth shapes called smooth manifolds and examines something called a hyperplane distribution. This distribution follows a special rule called 'complete non-integrability', which means it behaves in a unique way.

How Does it Connect to Other Math?

Contact geometry is related to another area of math called symplectic geometry. But while symplectic geometry works with shapes that have an even number of dimensions, contact geometry mainly studies shapes with an odd number of dimensions. Both of these areas are inspired by ideas from classical mechanics.

In mechanics, scientists study something called phase space, which shows all the possible states a system can be in. Contact geometry often looks at a special part of this space that has one fewer dimension.

Why is it Important?

This field helps mathematicians and scientists understand complicated systems and movements. It is useful in both pure math and practical physics.

Mathematical formulation

Contact geometry looks at special shapes on smooth surfaces in higher dimensions. It studies a special way to arrange "hyperplanes" — flat spaces that are one dimension smaller than the whole space — so they never line up in a simple 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 makes contact geometry special and useful.

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 twist along the y-axis. This example works for any R²ⁿ⁺¹ . It is standard because Darboux's theorem says 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. Each point in J¹(M,R) is a mapping from a small area 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 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 way to change one contact manifold to another while keeping their contact structure the same. This means that how lines and surfaces touch each other stays the same after the change.

There are different kinds of contact transformations. One important kind is called a strict contact transformation, which needs a special choice of contact forms to work. Another kind involves infinitesimal contact symmetries. These are connected to vector fields that create small changes in the contact structure. These ideas help mathematicians learn more about the properties of contact manifolds and their changes.

Submanifolds

In contact geometry, we study special types of submanifolds inside a contact manifold. These include contact submanifolds, which are submanifolds where a special condition is true, and isotropic submanifolds, where the tangent space at each point fits inside a certain area.

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 different.

Vector fields

Liouville

In a symplectic manifold, a vector field is called Liouville if it meets a special rule linked to the symplectic form. This rule helps us match the manifold to a standard one.

A Liouville form is a 1-form. When we change this form, it creates a symplectic form. The tautological 1-form is one example of a Liouville form.

Reeb

Main article: Reeb vector field

For a contact form on a manifold, there is a special Reeb vector field, also called a characteristic vector field. This vector field is unique. It shows that certain pairs of directions stay the same as the Reeb vector field moves.

The Reeb vector field is not part of the contact structure. It belongs to the contact dynamics. If a contact form comes from a constant-energy surface inside a symplectic manifold, the Reeb vector field is the part of the Hamiltonian vector field that relates to the energy function on that surface.

We can use the Reeb vector field to study the shape of the contact manifold or the manifold itself. We use tools like Floer homology, such as symplectic field theory and, in three dimensions, embedded contact homology. The Reeb vector field is named after Georges Reeb.

Relation with symplectic geometry

Contact geometry and symplectic geometry are connected in many ways, often inspired by physics. A symplectic form works with even dimensions, while a contact form works with odd 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 an 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, showing 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 others 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 for 3-dimensional manifolds. The Gray stability theorem shows that contact structures on closed manifolds cannot be changed into different structures through continuous movements.

History

The ideas behind contact geometry have been around for a long time. Early mathematicians like Apollonius of Perga, Christiaan Huygens, Isaac Barrow, and Isaac Newton had some of these ideas. Later, Sophus Lie developed the theory of contact transformations. He used it to study math problems and understand how space changes.

The term "contact manifold" was first used in a paper 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 how heat works in different materials.

Scientists use contact geometry to solve difficult math problems about shapes in space. They have discovered special properties of certain shapes and how knots behave in three dimensions. This shows how useful contact geometry is in both natural science and pure math.