On Eisenhart's type theorem for sub-Riemannian metrics on step $2$ distributions with $\mathrm{ad}$-surjective Tanaka symbols

Zaifeng Lin, Igor Zelenko

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) geod…