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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
Invariant Nijenhuis Tensors and Integrable Geodesic Flows
Konrad Lomperta, Andriy Panasyukb a Faculty of Mathematics and Information Science, Warsaw University of Technology,
ul. Koszykowa 75, 00-662 Warszawa, Poland
b Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. S loneczna 54, 10-710 Olsztyn, Poland
Аннотация:
We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $\Pi_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $\Pi_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of compact Lie groups $G$ for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of $G$ to two subgroups $G=G_1\cdot G_2$, where $G/G_i$ are symmetric spaces, $K=G_1\cap G_2$.
Ключевые слова:
bi-Hamiltonian structures, integrable systems, homogeneous spaces, Lie algebras, Liouville integrability.
Поступила: 19 декабря 2018 г.; в окончательном варианте 2 августа 2019 г.; опубликована 7 августа 2019 г.
Образец цитирования:
Konrad Lompert, Andriy Panasyuk, “Invariant Nijenhuis Tensors and Integrable Geodesic Flows”, SIGMA, 15 (2019), 056, 30 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1492 https://www.mathnet.ru/rus/sigma/v15/p56
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