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.