Igor Zelenko


Lin Z., Zelenko I.
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 $\mathrm{ad}$-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.
Keywords: sub-Riemannian geometry, Riemannian geometry, sub-Riemannian Geodesics, separation of variables, nilpotent approximation, Tanaka symbol, orbital equivalence, overdetermined PDEs, graded nilpotent Lie algebras
Citation: Lin Z., Zelenko I.,  On Eisenhart's Type Theorem for Sub-Riemannian Metrics on Step $2$ Distributions with $\mathrm{ad}$-Surjective Tanaka Symbols, Regular and Chaotic Dynamics, 2024, vol. 29, no. 2, pp. 304-343

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