Subbundle

In mathematics, a subbundle ${\displaystyle L}$of a vector bundle ${\displaystyle E}$over a topological space ${\displaystyle M}$is a collection of linear subspaces ${\displaystyle L_{x}}$of the fibers ${\displaystyle E_{x}}$of ${\displaystyle E}$at ${\displaystyle x}$in ${\displaystyle M,}$that make up a vector bundle in their 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 ${\displaystyle N_{x}}$of ${\displaystyle x\in M}$, a set of vector fields ${\displaystyle Y_{k}}$span the vector spaces ${\displaystyle L_{y},y\in N_{x},}$and all Lie commutators ${\displaystyle \left[Y_{i},Y_{j}\right]}$are linear combinations of ${\displaystyle Y_{1},\dots ,Y_{n}}$then one says that ${\displaystyle L}$is an involutive distribution.

• 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