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