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Encyclopedia > Subbundle

In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...


In connection with foliation theory, a subbundle of the tangent bundle of a smooth manifold may be called a distribution (of tangent vectors). In mathematics, informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a stripy fabric. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...


If a set of vector fields Yk span the vector space U, and all Lie commutators [Yi,Yj] are linear combinations of the Yk, then one says that U is an involutive distribution. In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. ...


See also

In mathematics, the Frobenius theorem states, in its smooth version of degree 1, the following: Let U be an open set in Rn and F a submodule of Ω1(U) of constant rank r in U. Then F is integrable if and only if for every p ∈ U the stalk... In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. ...

References


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Springer Online Reference Works (703 words)
The concept of a vector bundle arose as an extension of the tangent bundle and the normal bundle in differential geometry; by now it has become a basic tool for studies in various branches of mathematics: differential and algebraic topology, the theory of linear connections, algebraic geometry, the theory of (pseudo-) differential operators, etc.
is said to be a subbundle of the vector bundle
The concepts of a subbundle and a quotient bundle are used in contraction and glueing operations used to construct vector bundles over quotient spaces.
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