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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“The virtue of art lies in detachment, in sequestering one object from the embarrassing variety. Until one thing comes out from the connection of things, there can be enjoyment, contemplation, but no thought.”
—Ralph Waldo Emerson (1803–1882)
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