Ehresmann Connection

An Ehresmann connection on E is a smooth subbundle H of TE, called the horizontal bundle of the connection, which is complementary to V, in the sense that it defines a direct sum decomposition TE = H⊕V (Kolář, Michor & Slovák 1993). In more detail, the horizontal bundle has the following properties.

• For each point e ∈ E, H_e is a vector subspace of the tangent space T_eE to E at e, called the horizontal subspace of the connection at e.
• H_e depends smoothly on e.
• For each e ∈ E, H_e ∩ V_e = {0}.
• Any tangent vector in T_eE (for any e∈E) is the sum of a horizontal and vertical component, so that T_eE = H_e + V_e.

In more sophisticated terms, such an assignment of horizontal spaces satisfying these properties corresponds precisely to a smooth section of the jet bundle J1E → E.