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35 relations: Adjoint representation, Algebra homomorphism, Algebraic topology, André Weil, Characteristic class, Characteristic polynomial, Chern class, Classifying space, Closed and exact differential forms, Complex vector bundle, Connection (mathematics), Connection form, Cup product, Curvature, Curvature form, De Rham cohomology, Differentiable manifold, Differential geometry, Exterior covariant derivative, Fixed-point subring, Foundations of Differential Geometry, Generalized Gauss–Bonnet theorem, Holomorphic vector bundle, Lie algebra, Lie group, Mathematics, Pfaffian, Pontryagin class, Principal bundle, Riemann sphere, Ring homomorphism, Ring of polynomial functions, Shiing-Shen Chern, Topology, Vector bundle.
Adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.
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Algebra homomorphism
A homomorphism between two associative algebras, A and B, over a field (or commutative ring) K, is a function F\colon A\to B such that for all k in K and x, y in A,.
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Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
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André Weil
André Weil (6 May 1906 – 6 August 1998) was an influential French mathematician of the 20th century, known for his foundational work in number theory, algebraic geometry.
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Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.
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Characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.
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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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Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG.
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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 (dα.
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Complex vector bundle
In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.
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Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.
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Connection form
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.
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Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring.
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Curvature
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
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Curvature form
In differential geometry, the curvature form describes the curvature of a connection on a principal bundle.
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De Rham cohomology
In mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
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Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
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Exterior covariant derivative
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.
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Fixed-point subring
In algebra, the fixed-point subring R^f of an automorphism f of a ring R is the subring of the fixed points of f: More generally, if G is a group acting on R, then the subring of R: is called the fixed subring or, more traditionally, the ring of invariants.
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Foundations of Differential Geometry
Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu.
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Generalized Gauss–Bonnet theorem
In mathematics, the generalized Gauss–Bonnet theorem (also called Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss and Pierre Ossian Bonnet) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature.
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Holomorphic vector bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Pfaffian
In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the size of the matrix.
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Pontryagin class
In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes.
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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 of a space with a group.
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Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.
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Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.
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Ring of polynomial functions
In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring.
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Shiing-Shen Chern
Shiing-Shen Chern (October 26, 1911 – December 3, 2004) was a Chinese-American mathematician who made fundamental contributions to differential geometry and topology.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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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. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
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Chern-Weil, Chern-Weil homomorphism, Chern-Weil theory, Chern–Weil theory, Weil homomorphism.
References
[1] https://en.wikipedia.org/wiki/Chern–Weil_homomorphism
