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This article deals with the local sub-Riemannian geometry on ℜ3,
(D,g) where D is the distribution ker ω, ω being the Martinet
one-form : dz - ½y2dxand g is a Riemannian metric on D. We prove that we can take
g as a sum of squares adx2 + cd2. Then we analyze the flat case where a = c = 1. We parametrize
the set of geodesics using elliptic integrals. This allows to compute
the exponential mapping, the wave front, the conjugate and cut loci
and the sub-Riemannian sphere. A direct consequence...
In the recent paper [Yan, Z.: Existence of homogeneous geodesics on homogeneous Finsler spaces of odd dimension, Monatsh. Math. 182,1, 165–171 (2017)], it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. However, the proof contains a serious gap. The situation is a bit delicate, because the statement is correct. In the present paper, the incorrect part in this proof is indicated. Further, it is shown that homogeneous geodesics in homogeneous...
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