A complete answer to the Gaveau--Brockett problem

The note is dedicated to provide a satisfying and complete answer to the long-standing Gaveau–Brockett open problem. More precisely, we determine the exact formula of the Carnot–Carathéodory distance on arbitrary step-two groups. The basic idea of the proof is combining Varadhan’s formulas with the explicit expression for the associated heat kernel ph and the method of stationary phase. However, we have to introduce a number of original new methods, especially the usage of the concept of “Operator convexity”. Next, new integral expressions for ph by means of properties of Bessel functions will be presented. An unexpected direct proof for the well-known positivity of ph via its original integral formula, will play an important role. Furthermore, all normal geodesics joining the identity element o to any given g as well as the cut locus can be characterized on every step-two groups. Finally, the corresponding results in Riemannian geometry on step-two groups will be briefly presented as well.

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