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This argument is similar to that of [12]. Let M be a smooth Riemannian manifold of dimension n with a Riemannian metric g. Let X1 , X2 , . . , Xn be a local orthonormal frame of T (M ) in a local path U . And let ω 1 , ω 2 , . . , ω n be its dual. The differential d and its dual ϑ acting on Γ(∧p T ∗ (M )) are written as follows,using the Levi-Civita connection ∇ (see Appendix A of [14]): n n ϑ=− e(ω j )∇Xj , d= j=1 ı(Xj )∇Xj , j=1 where we use the following notation. Notation. e(ω j )ω = ω j ∧ ω, ı(Xj )ω(Y1 , .

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1. The Heisenberg group If we identify R2 with the complex plane C via R2 (x, y) ↔ z = x + iy ∈ C and let H = C × R, then H becomes a non-commutative group when equipped with the multiplication · given by 1 (z, t) · (w, s) = z + w, t + s + [z, w] , (z, t), (w, s) ∈ H, 4 where [z, w] is the symplectic form of z and w defined by [z, w] = 2 Im(zw). In fact, H is a unimodular Lie group on which the Haar measure is just the ordinary Lebesgue measure dz dt. This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

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2-step nilpotent Lie groups of higher rank by Samiou E.


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