Contact odd, symplectic even: where Klein's five dimensions sit on Book 4's dimension ladder
Book 4 climbs a dimension ladder: a contact 3-manifold (Chapter 3), the four-dimensional symplectisation (Chapter 4), a five-dimensional jet space (Chapter 5) and a sixth slot where time returns (Chapter 6). Kaluza and Klein supply a concrete, historically real five-dimensional space. This chapter sets the two side by side and says exactly where they agree and where they do not.
Standard contact geometry has a clean rhythm. A contact structure lives on an odd-dimensional manifold of dimension 2n+1. Multiply it by a line and you get its symplectisation, an even-dimensional symplectic manifold of dimension 2n+2, with symplectic form d(esα) (Book 4, Ch. 4 §4.4 writes this in the same spirit). So the ladder alternates: 3 (contact), 4 (symplectic), 5 (contact), 6 (symplectic).
Take four-dimensional spacetime with coordinates and an electromagnetic potential A, and add Klein's circle coordinate y. The one-form η = dy + κA gives dη = κF, and the previous chapter showed η ∧ dη ∧ dη = κ² dy ∧ F ∧ F. So the Kaluza-Klein space is contact exactly where F ∧ F ≠ 0.
A concrete case. Let the potential give a constant electric field E along one axis and a constant magnetic field B parallel to it. In coordinates (t, x, u, v) for spacetime, with u, v the plane perpendicular to the field, take A = E x dt + B u dv. Then F = E dx ∧ dt + B du ∧ dv, so F ∧ F = 2EB dx ∧ dt ∧ du ∧ dv, which is nowhere zero when E and B are both nonzero. The Reeb field of η = dy + A is ∂/∂y, the hidden circle.
The same structure is the standard one in a more familiar guise. The Darboux form on R⁵ is dz − p₁ dx₁ − p₂ dx₂, the contact form of the first-order jet space J¹(R², R). Writing p dx = d(px) − x dp and absorbing d(px) into the circle coordinate shows it equals dy + x₁ dp₁ + x₂ dp₂, a Kaluza-Klein form with field F = dx₁ ∧ dp₁ + dx₂ ∧ dp₂ and F ∧ F = 2 dx₁ ∧ dp₁ ∧ dx₂ ∧ dp₂ ≠ 0. So the genuine five-dimensional contact jet space is the Kaluza-Klein space of a constant field with F ∧ F ≠ 0.
Book 4's Chapter 5 uses J¹(R, R²), coordinates (x, y₁, y₂, p₁, p₂), with two forms α_i = dy_i − p_i dx. That object is five-dimensional, but the plane field ker α₁ ∩ ker α₂ has dimension three. A contact structure on a five-manifold is a four-dimensional plane field. So J¹(R, R²) with those two forms is an exterior differential system (a corank-2 Pfaffian system), not a contact manifold. Book 4 Chapter 5 now carries a correction box saying the same. The space that does fit the ladder's odd rungs is J¹(R², R), which is the Kaluza-Klein space above.
Book 4 Chapter 6 adds time as a sixth direction, with Galilean contact transformations. The symplectisation of the five-dimensional contact space is also six-dimensional, but its extra coordinate s is the scale of the contact form, not time. Both give six dimensions. I do not claim they are the same object. Chapter 6's “Galilean” example also failed its own invariance check and is now marked as corrected in place; the construction should be redone against the symplectisation before the two are identified.
In Book 4, time is promoted to a coordinate at rung 2: autonomisation turns a time-dependent system into an autonomous one on a space with one more coordinate. Grabowska and Grabowski (arXiv:2207.04484) use the same device for contact Hamiltonians, on P × T*R. In Kaluza-Klein theory, time is already one of the four base coordinates, and the new direction is the circle. The ladder and the Kaluza line therefore add a dimension for different reasons: one to make a system autonomous, the other to carry electric charge.