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Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory.
A version of the second (differential) Bianchi identity from Riemannian geometry holds for a connection on any vector bundle. Recall that a connection ∇ {\displaystyle \nabla } on a vector bundle E → M {\displaystyle E\to M} induces an endomorphism connection on End ( E ) {\displaystyle \operatorname {End} (E)} .
Development (differential geometry) connection form; Cartan connection. affine connection; conformal connection; projective connection; method of moving frames; Cartan's equivalence method; Vierbein, tetrad; Cartan connection applications; Einstein–Cartan theory; connection (vector bundle) connection (principal bundle) Ehresmann connection ...
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.It uses the techniques of single variable calculus, vector calculus, linear algebra and multilinear algebra.
The Grothendieck connection is a generalization of the Gauss–Manin connection constructed in a manner analogous to that in which the Ehresmann connection generalizes the Koszul connection. The construction itself must satisfy a requirement of geometric invariance , which may be regarded as the analog of covariance for a wider class of ...
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 that admits gauge symmetry.
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. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his ...
A commonly cited example is the Dwork construction of the Picard–Fuchs equation.Let (,,) be the elliptic curve + + =.Here, is a free parameter describing the curve; it is an element of the complex projective line (the family of hypersurfaces in dimensions of degree n, defined analogously, has been intensively studied in recent years, in connection with the modularity theorem and its ...
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