Igor Zelenko
Publications:
Lin Z., Zelenko I.
On Eisenhart's Type Theorem for SubRiemannian Metrics on Step $2$ Distributions with $\mathrm{ad}$Surjective Tanaka Symbols
2024, vol. 29, no. 2, pp. 304343
Abstract
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 subRiemannian metrics on step $2$ distributions with $\mathrm{ad}$surjective Tanaka symbols. The class of adsurjective step $2$ nilpotent Lie algebras contains a wellknown class of algebras of Htype as a very particular case.
