Geodesics as products of one‐parameter subgroups in compact lie groups and homogeneous spaces

Nikolaos Panagiotis Souris

Mathematische Nachrichten, 2023, vol. 296, issue 6, 2609-2625

Abstract: We study the geodesic equation for compact Lie groups G and homogeneous spaces G/H$G/H$, and we prove that the geodesics are orbits of products exp(tX1)⋯exp(tXN)$\exp (tX_1)\cdots \exp (tX_N)$ of one‐parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group SO(3)$SO(3)$, we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.

Date: 2023
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https://doi.org/10.1002/mana.202000282

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