Courant bracket, the Glossary

Table of Contents

1. 46 relations: Alan Weinstein, Almost complex manifold, Antisymmetric relation, Automorphism, Chern class, Circle bundle, Closed and exact differential forms, Cohomology, Complex conjugate, Complex dimension, Complex manifold, Complex number, Cotangent bundle, Courant algebroid, Differential form, Differential geometry, Dirac structure, Exterior derivative, Foliation, Generalized complex structure, Gerbe, Inner product space, Integrability conditions for differential systems, Interior product, Isotropic quadratic form, Jacobi identity, Kalb–Ramond field, Kähler manifold, Lie algebroid, Lie derivative, Linear subspace, Manifold, Mathematics, Nigel Hitchin, Poisson manifold, Principal bundle, Product rule, Quadratic form, Riemann curvature tensor, String theory, Symplectic geometry, Tangent bundle, Tensor product, Theodore James Courant, Vector bundle, Vector field.

2. Binary operations

Alan Weinstein

Alan David Weinstein (born 17 June 1943) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry.

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Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space.

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Antisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if there is no pair of distinct elements of X each of which is related by R to the other.

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Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles.

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Circle bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1.

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Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero, and an exact form is a differential form, α, that is the exterior derivative of another differential form β.

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Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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Complex dimension

In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety.

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Complex manifold

In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are holomorphic. Courant bracket and complex manifold are differential geometry.

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Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

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Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.

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Courant algebroid

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. Courant bracket and Courant algebroid are differential geometry.

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Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. Courant bracket and differential form are differential geometry.

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Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

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Dirac structure

In mathematics a Dirac structure is a geometric construction generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. Courant bracket and Dirac structure are differential geometry.

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Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

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Foliation

In mathematics (differential geometry), 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.

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Generalized complex structure

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. Courant bracket and generalized complex structure are differential geometry.

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Gerbe

In mathematics, a gerbe is a construct in homological algebra and topology.

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Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.

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Integrability conditions for differential systems

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms.

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Interior product

In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. Courant bracket and interior product are differential geometry.

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Isotropic quadratic form

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero.

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Jacobi identity

In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation.

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Kalb–Ramond field

In theoretical physics in general and string theory in particular, the Kalb–Ramond field (named after Michael Kalb and Pierre Ramond), also known as the Kalb–Ramond B-field or Kalb–Ramond NS–NS B-field, is a quantum field that transforms as a two-form, i.e., an antisymmetric tensor field with two indices.

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Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.

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Lie algebroid

In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. Courant bracket and Lie algebroid are differential geometry.

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Lie derivative

In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. Courant bracket and Lie derivative are differential geometry.

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Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspaceThe term linear subspace is sometimes used for referring to flats and affine subspaces.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Nigel Hitchin

Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics.

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Poisson manifold

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. Courant bracket and Poisson manifold are differential geometry.

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Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with. Courant bracket and principal bundle are differential geometry.

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Product rule

In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.

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Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial).

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Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. Courant bracket and Riemann curvature tensor are differential geometry.

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String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

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Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Courant bracket and symplectic geometry are differential geometry.

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Tangent bundle

A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself.

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Tensor product

In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted.

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Theodore James Courant

Theodore James "Ted" Courant is an American mathematician who has conducted research in the fields of differential geometry and classical mechanics.

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Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g.

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Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n.

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See also

Binary operations

• Absorbing element
• Barrel shifter
• Binary operation
• Blaschke sum
• Cap product
• Circular convolution
• Commutator
• Composition of relations
• Courant bracket
• Cup product
• DE-9IM
• Demonic composition
• Elvis operator
• Exponentiation
• Ext functor
• Function composition
• Icosian calculus
• Identity element
• Inverse element
• Iterated binary operation
• Join and meet
• Light's associativity test
• Logic alphabet
• Logical connectives
• Logical consequence
• Lulu smoothing
• Magma (algebra)
• Mean operation
• Minkowski addition
• Modular multiplicative inverse
• Null coalescing operator
• Operations on numbers
• Relational operator
• Tor functor
• Wreath product

References

[1] https://en.wikipedia.org/wiki/Courant_bracket

Also known as Dorfman bracket.