In mathematics, a subbundle L {\displaystyle L} of a vector bundle E {\displaystyle E} over a topological space M {\displaystyle M} is a subset of E {\displaystyle E} such that for each x {\displaystyle x} in M , {\displaystyle M,} the set L x {\displaystyle L_{x}} , the intersection of the fiber E x {\displaystyle E_{x}} with L {\displaystyle L} , is a vector subspace of the fiber E x {\displaystyle E_{x}} so that L {\displaystyle L} is a vector bundle over M {\displaystyle M} in its own right. In connection with foliation theory, a subbundle of the tangent bundle of a smooth manifold may be called a distribution (of tangent vectors). 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}} span the vector spaces L y , y ∈ N x , {\displaystyle L_{y},y\in N_{x},} and all Lie commutators [ 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 involutive distribution.
See also Frobenius theorem (differential topology) – On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs Sub-Riemannian manifold – Type of generalization of a Riemannian manifold
