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

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.