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32 relations: Élie Cartan, Chern–Simons form, Connection (principal bundle), Connection form, Curvature, Curvature of Riemannian manifolds, Differential form, Differential geometry, Ehresmann connection, Exterior algebra, Exterior covariant derivative, Exterior derivative, Flat vector bundle, Foundations of Differential Geometry, Frame bundle, Fundamental vector field, Gauge theory, Introduction to the mathematics of general relativity, John Wiley & Sons, Katsumi Nomizu, Lie algebra, Lie algebra-valued differential form, Lie group, Luigi Bianchi, Principal bundle, Riemann curvature tensor, Riemannian geometry, Riemannian manifold, Shoshichi Kobayashi, Skew-symmetric matrix, Solder form, Tangent bundle.
Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
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Chern–Simons form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes.
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Connection (principal bundle)
In mathematics, a connection 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.
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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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Curvature
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
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Curvature of Riemannian manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.
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Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
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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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Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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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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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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Flat vector bundle
In mathematics, a vector bundle is said to be flat if it is endowed with an linear connection with vanishing curvature, ie. a flat connection.
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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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Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex.
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Fundamental vector field
In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold.
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Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
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Introduction to the mathematics of general relativity
The mathematics of general relativity is complex.
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John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
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Katsumi Nomizu
was a Japanese-American mathematician known for his work in differential geometry.
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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 algebra-valued differential form
In differential geometry, a Lie algebra-valued form is a differential form with values in a Lie algebra.
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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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Luigi Bianchi
Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian mathematician.
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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 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 method used to express the curvature of Riemannian manifolds.
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Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Shoshichi Kobayashi
was a Japanese-American mathematician.
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Skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.
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Solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent.
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Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.
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Bianch's identity, Bianchi identities, Bianchi identity, Bianchi's identity, Bianchi's second identity, Curvature 2-form, Differential Bianchi identity, E. Cartan's structure equation, First Bianchi identity, Flat connection, Second Bianchi identity.
References
[1] https://en.wikipedia.org/wiki/Curvature_form
