Subbundle

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

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

References[]

• "Involutive Distribution". PlanetMath.^{[dead link]}