Subbundle

A subbundle $L$ of a vector bundle $E$ over a topological space $M$.

In mathematics, a subbundle $L$ of a vector bundle $E$ over a topological space $M$ is a subset of $E$ such that for each $x$ in $M,$ the set $L_{x}$, the intersection of the fiber $E_{x}$ with $L$, is a vector subspace of the fiber $E_{x}$ so that $L$ is a vector bundle over $M$ 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 $N_{x}$ of $x\in M$, a set of vector fields $Y_{k}$ span the vector spaces $L_{y},y\in N_{x},$ and all Lie commutators $\left[Y_{i},Y_{j}\right]$ are linear combinations of $Y_{1},\dots ,Y_{n}$ then one says that $L$ 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