Subbundle

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a subbundle 
 
 
 
 L
 
 
 {\displaystyle L}
 
 of a <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a> 
 
 
 
 E
 
 
 {\displaystyle E}
 
 over a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a collection of <a href="/facts/Linear_subspace/2wtKTgHL">linear subspaces</a> 
 
 
 
 
 L
 
 x
 
 
 
 
 {\displaystyle L_{x}}
 
of the fibers 
 
 
 
 
 E
 
 x
 
 
 
 
 {\displaystyle E_{x}}
 
 of 
 
 
 
 E
 
 
 {\displaystyle E}
 
 at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in 
 
 
 
 M
 ,
 
 
 {\displaystyle M,}
 
 that make up a vector bundle in their own right.
In connection with <a href="/facts/Foliation/cxml9vyJ">foliation</a> theory, a subbundle of the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> of a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> may be called a <a href="/facts/Distribution_(differential_geometry)/N3My5NLR">distribution</a> (of <a href="/facts/Tangent_vector/UEq5yffm">tangent vectors</a>). 
If locally, in a neighborhood 
 
 
 
 
 N
 
 x
 
 
 
 
 {\displaystyle N_{x}}
 
 of 
 
 
 
 x
 ∈
 M
 
 
 {\displaystyle x\in M}
 
, a set of vector fields 
 
 
 
 
 Y
 
 k
 
 
 
 
 {\displaystyle Y_{k}}
 
 <a href="/facts/Linear_span/vPjclEtb">span</a> the vector spaces 
 
 
 
 
 L
 
 y
 
 
 ,
 y
 ∈
 
 N
 
 x
 
 
 ,
 
 
 {\displaystyle L_{y},y\in N_{x},}
 
 and all <a href="/facts/Lie_commutator/1KSJvrzk">Lie commutators</a> 
 
 
 
 
 [
 
 
 Y
 
 i
 
 
 ,
 
 Y
 
 j
 
 
 
 ]
 
 
 
 {\displaystyle \left[Y_{i},Y_{j}\right]}
 
 are linear combinations of 
 
 
 
 
 Y
 
 1
 
 
 ,
 …
 ,
 
 Y
 
 n
 
 
 
 
 {\displaystyle Y_{1},\dots ,Y_{n}}
 
 then one says that 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is an <a href="/facts/Involutive_distribution/N3My5NLR">involutive distribution</a>.

Subbundle open-in-new

Subbundle