In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
In vector calculus, and more generally differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.,
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.
In mathematics, 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. If the fiber bundle is a vector bundle, then the notion of parallel transport must be linear. Such a connection is equivalently specified by a covariant derivative, which is an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Connections in this sense generalize, to arbitrary vector bundles, the concept of a linear connection on the tangent bundle of a smooth manifold, and are sometimes known as linear connections. Nonlinear connections are connections that are not necessarily linear in this sense.
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 abstract algebra and multilinear algebra, a multilinear form on is a map of the type
,
The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, zweibein, fünfbein, elfbein etc. have been used. Vielbein covers all dimensions.
In mathematics, the interior product is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product ιXω is sometimes written as X ⨼ ω.
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure.
In differential geometry, a Lie algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations.
In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold.
The Lee Hwa Chung theorem is a theorem in symplectic topology.
In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold , sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem. More precisely, the strong Lefschetz property asks that for , the cup product
In the theory of partial differential equations, Holmgren's uniqueness theorem, or simply Holmgren's theorem, named after the Swedish mathematician Erik Albert Holmgren (1873–1943), is a uniqueness result for linear partial differential equations with real analytic coefficients.
In the fields of computer vision and image analysis, the scale-invariant feature operator is an algorithm to detect local features in images. The algorithm was published by Förstner et al. in 2009.
In statistics, the matrix t-distribution is the generalization of the multivariate t-distribution from vectors to matrices. The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution. For example, the matrix t-distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse Wishart distribution.
Infinite derivative gravity is a theory of gravity which attempts to remove cosmological and black hole singularities by adding extra terms to the Einstein–Hilbert action, which weaken gravity at short distances.
This article summarizes important identities in exterior calculus.
