Subbundle information

In mathematics, a subbundle U {\displaystyle U} of a vector bundle V {\displaystyle V} on a topological space X {\displaystyle X} is a collection of linear...

Lie bracket of two sections of the holomorphic subbundle is another section of the holomorphic subbundle. In generalized complex geometry one is not interested...

taking subbundles of other vector bundles. Given a vector bundle π : E → X {\displaystyle \pi :E\to X} over a topological space, a subbundle is simply...

E {\displaystyle VE} and horizontal bundle H E {\displaystyle HE} are subbundles of the tangent bundle T E {\displaystyle TE} of E {\displaystyle E} whose...

with a preferred complex distribution L, or in other words a complex subbundle of the complexified tangent bundle C T M = T M ⊗ R C {\displaystyle \mathbb...

pushforward of tangent vectors. The horizontal spaces together form a vector subbundle of T E {\displaystyle TE} . This has the immediate benefit of being definable...

The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any norm on the horizontal subbundle gives...

map is given as follows: since X is compact, any vector bundle E is a subbundle of a trivial bundle: E ↪ X × R n + k {\displaystyle E\hookrightarrow X\times...

{\displaystyle \varphi } -invariant subbundles must first be defined. In Hitchin's original discussion, a rank-1 subbundle labelled L is φ {\displaystyle \varphi...

in the horizontal direction" (i.e., the horizontal bundle is a vector subbundle of the tangent bundle of the fiber bundle), even if they are not "linear...

tautological bundle. More precisely, this is called the tautological subbundle, and there is also a dual n-dimensional bundle called the tautological...

that E ⊕ E' is trivial. Choose F' in the same way. Then let E ⊗ F be the subbundle of (E ⊕ E') ⊗ (F ⊕ F') with the desired fibers. Finally, use the approximation...

Riemannian vector bundle E, the orthonormal frame bundle is a principal O(k)-subbundle of the general linear frame bundle. In other words, the inclusion map...

on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponentially contracting, and one that is exponentially...

{\mathcal {O}}_{X}} is a subsheaf but typically not a subbundle (since any line bundle has only two subbundles). The quasi-coherent sheaves on any fixed scheme...

cotangent bundles. In both cases one demands that the direct sum of the subbundle and its complex conjugate yield the original bundle. An almost complex...

called the tautological line bundle. It is equivalently defined as the subbundle of the product C n + 1 × C P n → C P n {\displaystyle \mathbf {C} ^{n+1}\times...

Riemannian symmetric space that is additionally equipped with a parallel subbundle of End(TM) isomorphic to the imaginary quaternions at each point, and...

that the (fiberwise) inverse image of the values of this section form a subbundle of P {\displaystyle P} that is a principal H {\displaystyle H} -bundle...

{\displaystyle \mathbf {Gr} (k,{\mathcal {E}})(T)} are exactly the projective subbundles of rank k {\displaystyle k} in P ( E ) × S T . {\displaystyle \mathbf...

)\subset TM\oplus T^{*}M} defines a Dirac structure, i.e. a Lagrangian subbundle D ⊂ T M ⊕ T ∗ M {\displaystyle D\subset TM\oplus T^{*}M} which is closed...

is a principal Aff(n)-bundle Q over M, together with a principal GL(n)-subbundle P of Q and a principal Aff(n)-connection α (a 1-form on Q with values...

complex structure is actually a complex structure precisely when these subbundles are involutive, i.e., closed under the Lie bracket of vector fields, and...

a smooth map f if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under f, with...