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 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.

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