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Riemannian geometry and elliptic operators are closely related. The most simple Riemannian metric is the flat one and the associated elliptic operator is the usual laplacian which is invariant under translations on the additive group (IR,+). In an analogus way, for the sub-Riemanian geometry, the associated operator is sub-elliptic(see [BGG1],[BGG2] ).Heisenberg group is the prototype for this geometry. It is the simplest non-commutative Lie group.

The left invariant metrics which is defined on Heisenberg group