@examples of contact structures
Introduction
Contact structures are an essential concept in the field of differential geometry, particularly in the study of contact topology. A contact structure on a manifold is a geometric structure that can be thought of as a maximally non-integrable hyperplane distribution. This article explores various examples of contact structures, delving into their properties, applications, and significance in mathematics and physics.
Basic Definitions
A contact structure on a (2n+1)-dimensional manifold M is a smooth hyperplane distribution ξ in the tangent bundle TM that is completely non-integrable. This means that locally, ξ can be defined as the kernel of a 1-form α such that α ∧ (dα)^n is nowhere zero. The pair (M, ξ) is called a contact manifold. The 1-form α is referred to as a contact form, and the condition α ∧ (dα)^n ≠ 0 is known as the contact condition.
Standard Contact Structures
Contact Structures on Euclidean Space
The simplest example of a contact structure is found on the (2n+1)-dimensional Euclidean space ℝ^{2n+1}. Consider ℝ^3 with coordinates (x, y, z). The standard contact structure is given by the 1-form α = dz - y dx. The distribution ξ = ker(α) defines a contact structure because α ∧ dα = dx ∧ dy ∧ dz, which is non-zero everywhere.
Contact Structures on the Sphere
Another fundamental example is the standard contact structure on the odd-dimensional sphere S^{2n+1}. This structure can be obtained by considering S^{2n+1} as the unit sphere in ℂ^{n+1} and taking the contact form α = i(∑_{j=1}^{n+1} (z_j d\bar{z}_j - \bar{z}_j dz_j)), where z_j are the complex coordinates. The kernel of this form defines a contact structure on S^{2n+1}.
Overtwisted and Tight Contact Structures
Contact structures can be classified into two types: overtwisted and tight. An overtwisted contact structure contains an embedded disk such that the contact planes are tangent to the boundary of the disk. In contrast, a tight contact structure does not contain such disks. The distinction between these types is crucial in the study of 3-manifolds.
Overtwisted Structures
Overtwisted contact structures are more flexible and easier to construct. They are classified by homotopy classes of plane fields. The existence of overtwisted structures on any closed, oriented 3-manifold was established by Yakov Eliashberg, who showed that every such manifold admits an overtwisted contact structure.
Tight Structures
Tight contact structures are more rigid and difficult to classify. They are of particular interest due to their connection with symplectic geometry. The classification of tight contact structures is a challenging problem and has been solved only for specific classes of 3-manifolds, such as lens spaces and certain Seifert fibered spaces.
Examples in Higher Dimensions
Contact structures are not limited to 3-dimensional manifolds. In higher dimensions, they exhibit more complex behaviors and interactions with other geometric structures.
Higher-Dimensional Spheres
For odd-dimensional spheres S^{2n+1}, the standard contact structure can be generalized from the 3-dimensional case. The contact form α = i(∑_{j=1}^{n+1} (z_j d\bar{z}_j - \bar{z}_j dz_j)) defines a contact structure on S^{2n+1} for any n.
Contact Structures on Projective Spaces
Contact structures can also be defined on complex projective spaces. For instance, the complex projective space ℂP^{2n+1} admits a natural contact structure. This structure is related to the Hopf fibration, where the fibers are contact submanifolds.
Applications and Significance
Contact structures have significant applications in various fields of mathematics and physics. They play a crucial role in the study of dynamical systems, where they provide a natural framework for understanding the behavior of certain types of flows. In physics, contact structures are used in the formulation of classical mechanics and thermodynamics, particularly in the context of Hamiltonian systems.