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ЖУРНАЛЫ // Regular and Chaotic Dynamics // Архив

Regul. Chaotic Dyn., 2024, том 29, выпуск 2, страницы 304–343 (Mi rcd1257)

On Eisenhart’s Type Theorem for Sub-Riemannian Metrics on Step $2$ Distributions with $\mathrm{ad}$-Surjective Tanaka Symbols

Zaifeng Lin, Igor Zelenko

Department of Mathematics, Texas A\&M University, TX 77843 College Station, USA

Аннотация: The classical result of Eisenhart states that, if a Riemannian metric $g$ admits a Riemannian metric that is not constantly proportional to $g$ and has the same (parameterized) geodesics as $g$ in a neighborhood of a given point, then $g$ is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step $2$ graded nilpotent Lie algebras, called \emph{$\mathrm{ad}$-surjective}, and extend the Eisenhart theorem to sub-Riemannian metrics on step $2$ distributions with $\mathrm{ad}$-surjective Tanaka symbols. The class of ad-surjective step $2$ nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case.

Ключевые слова: sub-Riemannian geometry, Riemannian geometry, sub-Riemannian Geodesics, separation of variables, nilpotent approximation, Tanaka symbol, orbital equivalence, overdetermined PDEs, graded nilpotent Lie algebras

MSC: 53C17, 58A30, 58E10, 53A15, 37J39, 35N10, 17B70

Поступила в редакцию: 05.09.2023
Принята в печать: 04.01.2024

Язык публикации: английский

DOI: 10.1134/S1560354724020023



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