In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension.

Mathematical formulation

Contact structure

Given an <math>n</math>-dimensional smooth manifold <math>M</math>, and a point <math>p \in M</math>, a contact element of <math>M</math> with contact point <math>p</math> is an <math>n-1</math>-dimensional linear subspace of the tangent space to <math>M</math> at <math>p</math>. A contact structure on an odd dimensional manifold <math>M</math>, of dimension <math>2n+1</math>, is a smooth distribution of contact elements, denoted by <math>\xi</math>, which is generic (in the sense of being maximally non-integrable) at each point. A contact manifold is a smooth manifold equipped with a contact structure.

Due to the ambiguity by multiplication with a nonzero smooth function, the space of all contact elements of <math>M</math> can be identified with a quotient of the cotangent bundle <math>T^*M</math> (with the zero section <math>0_M</math> removed), namely:

Another perspective is via the Lie algebra of the distribution. There exists up to <math> n</math> vector fields <math> v_1, \dots, v_n</math> in the distribution such that they do not generate .

Examples

The standard contact structure

thumb|right|The standard contact structure on R<sup>3</sup>, of the one-form

thumb|The standard contact structure is isomorphic to the cylindrically symmetric <math>dz - r^2 d\theta</math>.

The standard contact structure in <math>\R^3</math>, with coordinates (x,y,z), is the one-form The contact plane ξ at a point (x,y,z) is spanned by the vectors and

These planes appear to twist along the y-axis. It is not integrable, as can be verified by drawing an infinitesimal square in the x-y plane, and follow the path along the one-forms. The path would not return to the same z-coordinate after one circuit. This is an instance of the Chow–Rashevskii connectivity theorem.

This example generalizes to any <math>\R^{2n+1}</math>. Its standard contact structure is <math>\theta := dz - \Sigma_{i=1}^n y_i dx_i</math>. 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 <math>\mathbb S^{2n-1}</math>is obtained by restricting the Liouville 1-form <math>\lambda = \Sigma_i\left(x_i d y_i -y_i d x_i\right) </math> on <math>\R^{2n}</math> to the unit sphere. Equivalently, it is obtained by the Liouville 1-form on <math>\mathbb C^n</math> <math>\Sigma_j z_j d \bar{z}_j-\bar{z}_j d z_j = dr \circ J</math>, where <math>J</math> is the multiplication by <math>i</math>, i.e. the standard complex structure on <math>\mathbb C^n</math>.

The Reeb vector field is <math>\Sigma_{j=1}^n\left(x_j \partial_{y_j}+y_j \partial_{x_j}\right)=\Sigma_{j=1}^n\left(z_j \partial_{z_j}+\bar{z}_j \partial_{\bar{z}_j}\right)</math>, which generates the Hopf fibration.

Equivalently, consider the standard symplectic structure <math>\omega = \Sigma_i dx_i \wedge dy_i</math> on <math>\R^{2n}</math>. Each 1-dimensional subspace <math>V</math> is isotropic, and has a complementary coisotropic subspace <math>V^\omega</math> that contains it. Projectivized to <math>\mathbb P(\R^{2n})</math>, each point in <math>\mathbb P(\R^{2n})</math> has a complementary plane that contains the point. This distribution of planes is isomorphic to the standard contact structure on <math>\mathbb S^{2n-1}</math>.

One-jet

Given a manifold <math>M</math> of dimension <math>n</math>, the one-jet space <math>J^1(M, \R)</math> is the space of germs of type <math>M \to \R</math> identified up to order-1 contact. Intuitively, each point in <math>J^1(M, \R)</math> is a mapping from an infinitesimal neighborhood of <math>M</math> to <math>\R</math>. Each member of the space can be identified by the three quantities <math>x \in M, f(x) \in \R, \nabla f(x) \in T^*_x M</math>, thus <math>J^1(M, \R)</math> is a manifold of dimension <math>2n+1</math> and can be identified with <math>T^*M \times \R</math>. It has a natural contact form <math>\alpha = df - \theta</math> given by the tautological 1-form <math>\theta = \Sigma_{i=1}^n y_i dx_i</math>. The standard contact structure is the special case where <math>M = \R^n </math>.

Any first-differentiable function <math>M \to \R</math> then uniquely lifts to a Legendrian submanifold in <math>J^1(M, \R)</math>, and conversely, any Legendrian submanifold is the lift of a first-differentiable function <math>M \to \R</math>. Its projection to <math>M \times \R</math> is the graph of the function. This also shows that <math>J^1(M, \R)</math> embeds into the contact bundle of hyperplane elements <math>C_n(M \times \R)</math>, defined below.

Contact bundle of hyperplane elements

Given a manifold <math>M</math> of dimension <math>n+1</math>, its n-th contact bundle <math>C_n M</math> is the bundle of its dimension-n contact elements. More abstractly, it is the projectivized cotangent bundle <math>C_n(M) \cong \mathbb P (T^* M)</math>. Locally, expand <math>M</math> in coordinates as <math>q^0, \dots, q^n</math>, then the contact bundle locally has coordinates <math>(q^0, \dots, q^n, [p_0, \dots, p_n])</math>, where <math>p_0, \dots, p_n</math> uses projective coordinates. Any n-submanifold of <math>M</math> uniquely lifts to an n-submanifold of <math>C_n M</math>. Conversely, an n-submanifold of <math>C_n (M)</math> is a lift of an n-submanifold of <math>M</math> iff it annihilates the 1-form <math>\Sigma_{\mu=0}^n p_\mu dq^\mu</math>. On the subset where <math>p_0 \neq 0</math>, the condition becomes <math>dq^0 + \Sigma_{i=1}^n p_i dq^i</math>, which is the standard contact structure.

Similarly, the contact bundle of cooriented hyperplane elements <math>C_n(M)^+ \cong \mathbb S(T^* M)</math> is obtained by spherizing the cotangent bundle, i.e. quotienting only by <math>\R^+</math>.

The contact structure on <math>C_n(M) </math> can also be described coordinate-free. Define <math>\pi : C_n(M) \to M</math> to be the fiber projection that maps a hyperplane element to its base point. Then, for any <math>\xi \in C_n(M)</math>, a local tangent vector <math>v \in T_\xi C_1(M)</math> is a simultaneous translation of the base point and a rotation of the hyperplane element. Then <math>v</math> is in the hyper-hyperplane at <math>\xi</math> iff <math>\pi(v)</math> is in the hyperplane element of <math>\xi</math> itself. In other words, the <math>2n

</math>-dimensional hyper-hyperplane at <math>\xi</math> is spanned by translation of the base point within <math>\xi</math>, as well as rotation of the hyperplane element while keeping its base point unchanged. A circle in the plane lifts to a helix in <math>C_1(M)^+ </math>, but a double helix in <math>C_1(M) </math>.

Others

Until the 1950s, the only contact manifolds were the above ones, until Boothby and Wang in 1958 made a general construction via contactization. This result generalizes to any compact almost-contact manifold.

Contact transformation

A contact transformation (or contactomorphism) is a diffeomorphism between two contact manifolds that preserves their contact structure. A contact symmetry is a contact transformation from a contact manifold to itself.

Let <math>(M, \alpha)</math> and <math>(M', \alpha')</math> be two manifolds equipped with contact forms. A diffeomorphism <math>f : M \to M'</math> is a contact transformation iff there exists some <math>\tau: M \to \R</math> that is nowhere zero, such that <math>f^* \alpha' = \tau \alpha</math>. If <math>\tau = 1</math> then it is a strict contact transformation. Note that the concept of a strict contact transformation depends on a particular choice of contact forms, and there are inequivalent choices. Therefore, there is no "strict contact transformation" between contact structures, only between contact forms.

A strict infinitesimal contact symmetry on <math>(M, \alpha)</math> is a vector field <math>V</math> such that <math>\mathcal L_V \alpha = 0</math>, where <math>\mathcal L</math> is the Lie derivative. An infinitesimal contact symmetry is a vector field <math>V</math> on the contact manifold that generates a one-parameter family of contact symmetries. Equivalently, if the hyperplane distribution is <math>\ker \alpha</math>, then the condition is <math>\mathcal L_V \alpha = \tau \alpha</math> for some <math>\tau: M \to \R</math>.

Similarly, the line-sphere correspondence and other transformations of the Lie sphere geometry are contact transformations. While a line has <math>\infty^1</math> points and a sphere has <math>\infty^2</math> points, they both have <math>\infty^2</math> infinitesimal planes. They were in fact some of the earliest ones considered by Lie.

Legendre transformation

Given <math>\R^{2n + 1}</math> with the standard contact structure, define its coordinates <math>(W, q^1, \dots, q^n, p_1, \dots, p_n)</math> such that the contact form is <math>dW - p_i dq^i</math>, then the Legendre transformation <math>(W, q, p) \mapsto (W - p_iq^i, p, -q)</math> is a strict contact transformation. It is obtained by contact-lifting the linear symplectic rotation <math>(q, p) \mapsto (p, -q)</math> of the symplectic space. This rotation is simply multiply-by-i of the standard linear complex structure on the symplectic space. In the plane, it exchanges a curve and its dual.

Since a differentiable function <math>F: \R^n \to \R </math> can be lifted uniquely to a Legendrian submanifold, and any contactomorphism preserves Legendrian submanifolds, this defines a Legendre transformation on the function <math>F </math> itself.

More generally, any differentiable real-valued function on any manifold <math>M </math> can be transformed using any contactomorphism on the one-jet space <math>J^1(M, \R)</math>. In particular, this defines the Legendre transformation for any manifold.

Canonical transformation

Given a manifold <math>M</math> with coordinates <math>(q^1, \dots, q^n)</math>, let <math>\theta = p_i dq^i</math> be the tautological one-form on its phase space <math>P = T^* M</math>, and let <math>\omega = dp_i \wedge dq^i = d\theta</math> be the symplectic form on the phase space. Extend by one dimension to <math>\R\times P</math> with coordinates <math>(W, q^1, \dots, q^n, p_1, \dots, p_n) </math>, then we have a contact manifold with the contact form <math>dW - \theta </math>. This can be interpreted as a lift of the Hamilton–Jacobi equation in time-independent Hamiltonian dynamics, with <math>W</math> being Hamilton's characteristic function. A canonical transformation <math>\Phi: P \to P </math> generated by <math>F: P \to \R</math> satisfies <math>\Phi^* \theta = \theta + dF</math>, and it lifts to a contact transformation <math>\hat \Phi : \R \times P \to \R \times P</math> by <math>\hat \Phi(W, q, p) = (W + F(q, p), \Phi(q, p)) </math>.

Others

Given any contact form, its corresponding Reeb vector field is a strict infinitesimal contact symmetry, and the Reeb flow is a one-parameter family of contact symmetries. The codeodesic flow is one example.

For the standard contact form on an odd-dimensional sphere, its Reeb flow generates its Hopf fibration.

Submanifolds

Contact

Given a contact manifold <math>(M, \alpha)</math>, a contact submanifold is some submanifold <math>L \subset M</math> such that <math>(L, \alpha|_L)</math> is a contact submanifold.

Isotropic

Given a contact manifold <math>(M, \alpha)</math>, an isotropic submanifold (or integral submanifold) is some submanifold <math>L \subset M</math> such that for any point <math>p \in L</math>, the tangent space is within the distribution <math>T_p L \subset \ker\alpha</math>, that is, <math>\alpha|_L = 0</math>.

In particular, since <math>(d\alpha)^n \wedge \alpha \neq 0</math>, at any point <math>p \in L</math>, <math>d\alpha_p </math> is a symplectic form on the hyperplane at <math>p </math>. Yet, we must also have <math>d\alpha|_L = 0 </math>, so <math>T_p L </math> is a null space in the local hyperplane, which must have dimension at most <math>n </math>.

Legendrian

As described above, an integral manifold can have up to n dimensions. These extremal integral manifolds are Legendrian submanifolds. Indeed, such submanifolds are extremely common, since they satisfy an h-principle:<blockquote>Given a symplectic manifold <math> (P, \omega)</math> and a compact and contact type <math> M \subset P</math>, construct a contact manifold <math> (M, \alpha|_M)</math> as described, then construct a positive (since <math> M</math> is coorientable) symplectization <math> (P^+, \omega')</math> where <math> P^+ = M \times \R^+</math>. Then there exists a neighborhood of <math> M \subset P</math> and a neighborhood of <math> M\times \{1\} \subset P^+</math> that are symplectically isomorphic.</blockquote>

Reeb transversal construction

Given a contact manifold <math>(M, \alpha)

</math>, construct local Darboux coordinates so that <math>\alpha = dW - \theta</math>, with <math>\theta = p_i dq^i </math>, then <math>d\alpha = -d \theta = \omega </math>, where <math>\omega = dq^i \wedge dp_i </math>, and the Reeb vector field <math>R = \partial_W </math>. Thus, if <math>P \subset M </math> is any <math>2n</math>-submanifold that is transverse to the Reeb vector field, then <math>(P, d\alpha|_P) </math> is a symplectic manifold. The Reeb vector field flow provides symplectomorphic homotopy between these, another instance of the h-principle.

Symplectization

Given any contact manifold <math>M</math> of dimension <math>2n-1</math> with a distribution of hyperplanes <math>\xi</math>, it can be symplectized to a symplectic manifold <math>(P, \omega)</math> of dimension <math>2n</math>. The manifold consists of covectors of <math>M</math> that are in full contact with the distribution of hyperplanes:<math display="block">P := \{(p, w) : p \in M, w \in T_p^* M, \ker w \in \xi\}</math>This produces <math>\theta</math>, a global tautological 1-form on <math>P</math>. Any vector <math>V \in T_{(p, w)}P</math> projects down to a vector <math>v \in T_pM</math>, and we define <math>\theta(V) := w(v)</math>. Then define <math>\omega := d\theta</math>. This is a symplectic form, as can be verified by constructing local Darboux coordinates. For example, given an n-manifold <math>M</math>, its contact bundle <math>C_{n-1}(M)</math> symplectizes to <math>T^*M \setminus \{0\}</math>, the nonzero cotangent bundle.

This construction does not depend on the choice of contact form. If a contact form <math>\alpha</math> were locally chosen, then <math display="block">P := \{(p, r\alpha_p) : p \in M, r \in \R \setminus\{0\}\}</math>and <math>\omega = d(r\alpha)</math>. <math>P</math> is a fiber bundle over <math>M</math>, with fibers being <math>\R \setminus \{0\}</math>. If the contact structure is coorientable, then the contact form can be chosen globally, and the fiber bundle splits into two trivial line bundles:<math display="block">P^\pm := \{(p, r\alpha_p) : p \in M, \pm r > 0\} \cong M \times \R</math>There is a bijection between 1-homogeneous infinitesimal symplectomorphisms of the symplectic manifold and infinitesimal contactomorphisms of the contact manifold. In one direction, given a vector field <math>v</math> on <math>M</math> that is an infinitesimal contactomorphism, it flows any <math>(p, w) \in P</math> to some <math>(p', w')</math>. Since it preserves the contact structure, <math>(p', w') \in P </math>. Further, for any <math>k \in \R \setminus \{0\} </math>, it flows <math>(p, kw)</math> to <math>(p', kw')</math>. Thus it lifts to a vector field <math>V</math> on <math>P</math>. This is an infinitesimal symplectomorphism that is 1-homogeneous along the fibers. Conversely, any infinitesimal symplectomorphism that is 1-homogeneous along the fibers projects down to infinitesimal contactomorphism.

Say that a Hamiltonian <math>H : P \to \R</math> is 1-homogeneous iff <math display="block">H(p, kw) = k H(p, w), \quad \forall (p, w) \in P, \; k \in \R\setminus\{0\}</math>then every infinitesimal contactomorphism of <math>P</math> is the projection of a Hamiltonian flow of <math>P</math> generated by some 1-homogeneous Hamiltonian.

Applications

Like symplectic geometry, contact geometry has broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, integrable systems and to control theory. Contact geometry also has applications to low-dimensional topology; for example, it has been used by Kronheimer and Mrowka to prove the property P conjecture, by Michael Hutchings to define an invariant of smooth three-manifolds, and by Lenhard Ng to define invariants of knots. It was also used by Yakov Eliashberg to derive a topological characterization of Stein manifolds of dimension at least six.

Contact geometry has been used to describe the visual cortex.

Partial differential equations

The original motivation for the study of contact geometry was in solving first-order partial differential equations (PDE). In general, the problem is finding some <math> z(x_1, \dots, x_n)</math> satisfying a PDE<math display="block"> F(x_1, \dots, x_n, \partial_1 z, \dots, \partial_n z, z) = 0</math>Sophus Lie's idea was to lift the equation to 1-jet space <math> J^1(\R^n, \R)</math>, in which the equation <math> F(x, y, z) = 0</math> specifies a 2n-dimensional hypersurface, and the problem reduces to finding Legendrian submanifolds within this hypersurface.

Geometric optics

thumb|Along a constant-speed geodesic curve, the unit velocity vector is transported, creating the geodesic flow on the unit tangent bundle. Dually, the unit co-vector is also transported, creating the cogeodesic flow on the unit cotangent bundle. The (co)geodesic flow is a special case of the Reeb flow.

The Huygens–Fresnel principle of wave propagation can be formalized as a contact transformation. Specifically, given a Riemannian n-manifold <math>M</math>, consider its unit-speed geodesic curves (i.e. parameterized by arc length). This produces a transport of unit-length tangent vectors, and thus a vector flow field on the unit tangent bundle <math>UT(M)</math>. This is the geodesic flow. Dually, the propagation of infinitesimal wavefronts (wavelets) produces a transport of unit-length cotangent vectors, and thus a vector flow field on the unit cotangent bundle <math>UT^*(M)</math>. This is the cogeodesic flow. The tautological 1-form on <math>T^*M</math> restricted to <math>UT^*(M)</math> is a contact form, which then induces a contact in <math>UT(M)</math>. The Huygens–Fresnel principle states that the (co)geodesic flow is a strict infinitesimal contact symmetry, and more precisely, it is the Reeb vector field.

Legendrian submanifolds in <math>UT^*(M)</math> correspond to wavefront surfaces in <math> M</math>, and wave propagation over time corresponds to applying Reeb flow to the wavefront Legendrian submanifold. Legendrian submanifolds in <math>UT(M)</math> correspond to special types of pencils of rays, and Reeb flow corresponds to ray propagation over time. That Reeb flow preserves Legendrian submanifolds implies the Malus–Dupin theorem. In particular, a single point source can be regarded either as a sphere of exiting rays, or a sphere of exiting wavefronts. They are both maximally extended compact Legendrian submanifolds.

thumb|Tangents and involutes of the cubic curve <math>y = x^3 </math>.

For example, wave propagation in the plane at constant speed is particularly simple, and becomes a helical shearing in <math>UT^*(\R^2) \cong \R^2 \times \mathbb S^1 </math>. Circular wavefronts exiting a single point in the plane is lifted to a helix exiting a single line in <math>\R^2 \times \mathbb S^1</math>. Given an involute of an evolute, the other involutes are obtained by the one-parameter family of contact transformations.

See also

  • Floer homology, some flavors of which give invariants of contact manifolds and their Legendrian submanifolds
  • Sub-Riemannian geometry
  • Contact bundle

References

Introductions to contact geometry

Applications to differential equations

Contact three-manifolds and Legendrian knots

Information on the history of contact geometry

  • Contact geometry Theme on arxiv.org
  • Contact manifold at the Manifold Atlas