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


  • 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.

author list (cited authors)

  • Lin, Z., & Zelenko, I.

complete list of authors

  • Lin, Zaifeng||Zelenko, Igor

Book Title

  • arXiv

publication date

  • August 2023