In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle. The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle is then a particular choice of a subbundle of tangent bundle which is complementary to vertical bundle.
More precisely, if Ï€ : E → M is a smooth fiber bundle over a smooth manifold M and e ∈ E with Ï€(e) = x ∈ M, then the vertical space VeE at e is the tangent space Te(Ex) to the fiber Ex containing e...