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

Zaifeng Lin, Igor Zelenko

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 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.

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

Abstract

The classical result of Eisenhart states that if a Riemannian metric admits a Riemannian metric that is not constantly proportional to and has the same (parameterized) geodesics as in a neighborhood of a given point, then is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step graded nilpotent Lie algebras, called -surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step 2 distributions with -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.
Paper Structure (17 sections, 36 theorems, 219 equations)

This paper contains 17 sections, 36 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Affine equivalence in Riemannian geometry: nonrigidity and product structure
  3. Affine equivalence of sub-Riemannian metrics: the main conjecture
  4. The role of Tanaka symbol/nilpotent approximation and the main result
  5. Direct product structure on the level of Tanaka symbol/nilpotent approximation
  6. Ad-surjective Tanaka symbols and the main result
  7. Orbital equivalence and fundamental algebraic system
  8. Proof of Theorem \ref{['mainthm']}
  9. Main steps in the proof of Theorem \ref{['mainthm']}
  10. Proof of Proposition \ref{['step1_lemma1']}
  11. Proof that $[D_r, D_t]\in \mathcal{L}_{X_{n_{r-1}+1}}+\mathcal{L}_{X_{n_{t-1}+1}} \quad \mathrm{mod}\,\, D_r+D_t\,\, r\neq t$ (toward the proof of Proposition \ref{['Dij^2_involutive']})
  12. Proof of Proposition \ref{['step1_lemma2']}
  13. Proof of Proposition \ref{['PropD2D3']}
  14. Completing the proof of Proposition \ref{['Dij^2_involutive']}
  15. Completing the proof of Theorem \ref{['mainthm']} by rotating the frames on $D_i$'s
  16. ...and 2 more sections

Key Result

Theorem 1.1

If a Riemannian metric $g$ is not affinely rigid near a point $q_0$, i.e., admits a locally affinely equivalent non-constantly proportional Riemannian metric in a neighborhood of a point $q_0$, then the metric $g$ is the direct product of two Riemannian metrics in a neighborhood of $q_0$.

Theorems & Definitions (80)

  • Theorem 1.1: Eisenhart1923
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Conjecture 1.7: jmz2019
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 70 more