 | This article does not cite any sources .(December 2020) (Learn how and when to remove this template message) |
In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
| | This article does not cite any sources .(December 2020) (Learn how and when to remove this template message) |
In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Let be a contact manifold, and let . Consider the set
of all nonzero 1-forms at , which have the contact plane as their kernel. The union
is a symplectic submanifold of the cotangent bundle of , and thus possesses a natural symplectic structure.
The projection supplies the symplectization with the structure of a principal bundle over with structure group .
When the contact structure is cooriented by means of a contact form , there is another version of symplectization, in which only forms giving the same coorientation to as are considered:
Note that is coorientable if and only if the bundle is trivial. Any section of this bundle is a coorienting form for the contact structure.
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fibre bundles.
In mathematics, a base or basis for the topology τ of a topological space (X, τ) is a family B of open subsets of X such that every open set is equal to a union of some sub-family of B. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.
In mathematics, the adele ring of a global field is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field, and is an example of a self-dual topological ring.
In mathematics, a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1. The number p is called the dimension of the foliation and q = n − p is called its codimension.
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.
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 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.
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded relative to the limit ordinal. The name club is a contraction of "closed and unbounded".
In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray is required to be homogeneous in the sense that its integral curves t→ΦHt(ξ)∈TM obey the rule ΦHt(λξ)=ΦHλt(ξ) in positive reparameterizations. If this requirement is dropped, H is called a semispray.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).
In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were introduced by Nigel Hitchin in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.
In a field of mathematics known as differential geometry, a Courant geometry was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990 the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on , called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids.
In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of a smooth manifold M . A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written πTM.
Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold . For instance, these are gauge theory of dislocations in continuous media when , the generalization of metric-affine gravitation theory when is a world manifold and, in particular, gauge theory of the fifth force.
In differential geometry, the integration along fibers of a k-form yields a -form where m is the dimension of the fiber, via "integration".
K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality of the policy in inventory control theory. The policy is characterized by two numbers s and S, , such that when the inventory level falls below level s, an order is issued for a quantity that brings the inventory up to level S, and nothing is ordered otherwise. Gallego and Sethi have generalized the concept of K-convexity to higher dimensional Euclidean spaces.
Martin Hairer's theory of regularity structures provides a framework for studying a large class of subcritical parabolic stochastic partial differential equations arising from quantum field theory. The framework covers the Kardar–Parisi–Zhang equation, the equation and the parabolic Anderson model, all of which require renormalization in order to have a well-defined notion of solution.
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a physical model of some natural phenomenon.
