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In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport.
Let M be a differentiable manifold, such as Euclidean space.A vector-valued function can be viewed as a section of the trivial vector bundle. One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on M.
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 ...
In the first sub-case, all of the torsion can be uniquely absorbed into the connection form. (Riemannian manifolds are an example, since the Levi-Civita connection absorbs all of the torsion). The connection coefficients and their invariant derivatives form a complete set of invariants of the structure, and the equivalence problem is solved.
Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics.
Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle E → X {\displaystyle E\to X} written as a Koszul connection on the C ∞ ( X ) {\displaystyle C^{\infty }(X ...
Then a principal-connection on is a differential 1-form on with values in the Lie algebra of which is -equivariant and reproduces the Lie algebra generators of the fundamental vector fields on . In other words, it is an element ω of Ω 1 ( P , g ) ≅ C ∞ ( P , T ∗ P ⊗ g ) {\displaystyle \Omega ^{1}(P,{\mathfrak {g}})\cong C^{\infty }(P ...
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection , in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form .
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