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arXiv · 2112.07822

The Carnot-Carath\'eodory distance on $2$-step groups

Hong-Quan Li
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Abstract

Combining Varadhan's formula, Loewner's theorem with the method of stationary phase, we study the exact formula of the Carnot-Carath\'eodory distance on $2$-step groups. The method is also adapted to determine all normal geodesics from the identity element to any given point (up to a set of measure zero). As an application, we characterize the squared sub-Riemannian distance as well as the cut locus on generalized Heisenberg-type groups and on star graphs respectively. Furthermore, the long-standing open problem of Gaveau-Brockett is completely solved in the case of $N_{3, 2}$, the free Carnot group of step two and 3 generators.

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Hong-Quan Li. 2021-12-15. The Carnot-Carath\'eodory distance on $2$-step groups. https://arxiv.org/abs/2112.07822

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