By Heinrich W. Guggenheimer

This article comprises an easy advent to non-stop teams and differential invariants; an in depth therapy of teams of motions in euclidean, affine, and riemannian geometry; extra. contains workouts and sixty two figures.

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A flow of symplectomorphisms t → φt with compact support in A is Hamiltonian if it is the projection onto R2n of a flow in Hom(H(n), vol, Lip)(A). Consider such a flow which joins identity id with φ. Take the lift of the flow t ∈ [0, 1] → φ˜t ∈ Hom(H(n), vol, Lip)(A). 21) 0 Proof. The curve is not horizontal. The tangent has a vertical part (equal to the Hamiltonian). 21). 20) is non-degenerate. For given φ with compact support in A and generating function F , look at the one parameter family: φ˜a (x, x¯) = (φ(x), x¯ + F (x) + a) The volume of the cylinder {(x, z) : x ∈ A, z between 0 and F (x) + a} attains the minimum V (φ, A) for an a0 ∈ R.

E. x ∈ R2n the curve t → φt (x) ∈ R2n is (locally) Lipschitz. e. curve t → φt (x) lifts to a horizontal curve t → φht (x, 0). 4. 6. 11, which comes after. 10 Let t → φ˜t ∈ Hom(H(n), vol, Lip) be a curve such that the function t → Φ(˜ x, t) = (φ˜t (˜ x), t) is locally Lipschitz from H(n) × R to itself. Then t → φ˜t is a constant curve. 43 3 THE HEISENBERG GROUP Proof. 15 for the group H(n) × R we obtain that Φ is almost everywhere derivable. 11 to deduce the claim. 11. Let N be a noncommutative Carnot group.

33 Let N , M be Carnot groups. We write N ≤ M if there is an injective group morphism f : N → M which commutes with dilatations. 34 Let X,Y be Lipschitz manifolds over the Carnot groups M, respectively N. If N ≤ M then there is no bi-Lipschitz embedding of X in Y . Rigidity in the sense of this section manifests in subtler ways. The purpose of Pansu paper [25] was to extend a result of Mostow [22], called Mostow rigidity. Although it is straightforward now to explain what Mostow rigidity means and how it can be proven, it is beyond the purposes of these notes.