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--Carath\'eodory 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 $p_h$ 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 $p_h$ by means of properties of Bessel functions will be presented. An unexpected direct proof for the well-known positivity of $p_h$ 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.