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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.
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.
An Ehresmann connection drops the differential operator completely and defines a connection axiomatically in terms of the sections parallel in each direction (Ehresmann 1950). Specifically, an Ehresmann connection singles out a vector subspace of each tangent space to the total space of the fiber bundle, called the horizontal space .
In mathematics, and especially differential geometry and gauge theory, 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.
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 ...
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 is torsion-free, i.e., T ∇ is zero, so that ∇ X Y − ∇ Y X = [X, Y]; parallel transport is an isometry, i.e., the inner products (defined using g) between tangent vectors are preserved. This connection is called the Levi-Civita connection. The term "symmetric" is often used instead of torsion-free for the first property.
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant use ...