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Discrete Geodesic Flows on Stiefel Manifolds

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Abstract

We study integrable discretizations of geodesic flows of Euclidean metrics on the cotangent bundles of the Stiefel manifolds \(V_{n,r}\). In particular, for \(n=3\) and \(r=2\), after the identification \(V_{3,2}\cong\mathrm{SO}(3)\), we obtain a discrete analog of the Euler case of the rigid body motion corresponding to the inertia operator \(I=(1,1,2)\). In addition, billiard-type mappings are considered; one of them turns out to be the “square root” of the discrete Neumann system on \(V_{n,r}\).

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Notes

  1. Here we use the identifications \( \mathrm{so} (n)\cong \mathrm{so} (n)^*\) and \( \mathrm{so} (r)\cong \mathrm{so} (r)^*\) induced by the invariant scalar product \(\langle \eta_1,\eta_2\rangle=-(1/2) \operatorname{tr} (\eta_1 \eta_2)\).

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Funding

The research of B. Jovanović was supported by the Serbian Ministry of Education, Science and Technological Development through the Mathematical Institute of the Serbian Academy of Sciences and Arts. The research of Yu. N. Fedorov was partially funded by the Spanish MINECO-FEDER grants MTM2016-80276-P and PGC2018-098676-B-I00/AEI/FEDER/UE and the provincial grant 2017SGR1049.

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Authors and Affiliations

  1. Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Božidar Jovanović

  2. Department of Mathematics, Polytechnic University of Catalonia, C. Pau Gargallo 14, 08028, Barcelona, Spain

    Yuri N. Fedorov

Authors
  1. Božidar Jovanović
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  2. Yuri N. Fedorov
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Correspondence to Božidar Jovanović.

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To Academician Valery Vasil’evich Kozlov on the occasion of his 70th birthday

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Jovanović, B., Fedorov, Y.N. Discrete Geodesic Flows on Stiefel Manifolds. Proc. Steklov Inst. Math. 310, 163–174 (2020). https://doi.org/10.1134/S0081543820050132

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