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Local connectedness is, by definition, a local property of topological spaces, i.e., a topological property P such that a space X possesses property P if and only if each point x in X admits a neighborhood base of sets that have property P. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
A local trivialisation for is equivalently given by a local section : and the connection one-form and curvature can be pulled back along this smooth map. This gives the local connection one-form A α = s α ∗ ν ∈ Ω 1 ( U α , ad ( P ) ) {\displaystyle A_{\alpha }=s_{\alpha }^{*}\nu \in \Omega ^{1}(U_{\alpha },\operatorname {ad} (P ...
In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman ...
An example of this is the function block diagram, one of five programming languages defined in part 3 of the IEC 61131 (see IEC 61131-3) standard that is highly formalized (see formal system), with strict rules for how diagrams are to be built. Directed lines are used to connect input variables to block inputs, and block outputs to output ...
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.
The connection on the frame bundle can also be described using K-invariant differential 1-forms on F. [7] [34] The orthonormal frame bundle F is a 3-manifold. One of the key facts about F is that it is (absolutely or completely) parallelizable, i.e. for n = dim F, there are n vector fields on F which form a basis at each point.
[11]: 98 Adding a "locality" modifier, that the results of two spatially well-separated measurements cannot causally affect each other, [5] does not make the combination relate to Bell's proof; the only interpretation that Bell assumed was the one he called local causality.
which vanishes if and only if Γ i kj is symmetric on its lower indices. Given a metric connection with torsion, one can always find a single, unique connection that is torsion-free, this is the Levi-Civita connection. The difference between a Riemannian connection and its associated Levi-Civita connection is the contorsion tensor.