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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.
Levi-Civita connection; parallel transport. 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 ...
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.
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 ...
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
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.
Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publishers. Both were published again in 1996 as Wiley Classics Library.
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 ...