Subbundle

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

In mathematics, a subbundle $U$ of a vector bundle $V$ on a topological space $X$ is a collection of linear subspaces $U_{x}$of the fibers $V_{x}$ of $V$ at $x$ in $X,$ 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 a set of vector fields $Y_{k}$ span the vector space $U,$ and all Lie commutators $\left[Y_{i},Y_{j}\right]$ are linear combinations of the $Y_{k},$ then one says that $U$ 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