Subbundle

A subbundle ${\displaystyle L}$ of a vector bundle ${\displaystyle E}$ over a topological space ${\displaystyle M}$.

In mathematics, a subbundle${\displaystyle L}$ of a vector bundle ${\displaystyle E}$ over a topological space ${\displaystyle M}$ is a subset of ${\displaystyle E}$ such that for each ${\displaystyle x}$ in ${\displaystyle M,}$ the set ${\displaystyle L_{x}}$, the intersection of the fiber ${\displaystyle E_{x}}$ with ${\displaystyle L}$, is a vector subspace of the fiber ${\displaystyle E_{x}}$ so that ${\displaystyle L}$ is a vector bundle over ${\displaystyle M}$ 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 ${\displaystyle N_{x}}$ 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.

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