Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Carnot-Carath\'eodory distance on $2$-step groups   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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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol