Levi-Civita's type theorem for projectively equivalent sub-Riemannian metrics on step 2 distributions

Abstract

This paper extends the classical results of Eisenhart (1923) and Levi-Civita (1896) on the classification of pairs of germs of Riemannian metrics with the same geodesics, either as parametrized curves (affine equivalence) or as unparametrized curves (projective equivalence), to sub-Riemannian metrics on arbitrary step 2 distributions: at every stable point in the projective case and at every point in the affine case. This extends the results of Zelenko (2004) for contact and even-contact distributions and the affine result of Lin and Zelenko (2024) for step 2 distributions with ad-surjective Tanaka symbols; our projective result covers the ad-surjective case as well. In the affine case, we prove that if a sub-Riemannian metric admits an affinely equivalent metric that is not constantly proportional to it, then locally it is a product of two sub-Riemannian metrics. In the projective case, we prove the 2019 conjecture of Jean, Maslovskaya, and Zelenko on generalized Levi-Civita pairs; in particular, the distribution admits a product structure, the squares of its components are involutive, and the restrictions of the two metrics to each leaf of the corresponding foliation are conformal to each other under the appropriate pullback. The proof is based on a new graph-theoretic formalism, in which compatible moving frames are represented by the vertices and edges of certain graphs. As a byproduct, we introduce a new discrete invariant of step 2 nilpotent graded Lie algebras, the characteristic tuple. The compatible graphs are used on the open dense set where this tuple is locally constant; the resulting bracket relations are identities between smooth tensors, so they extend by continuity across the points where the tuple jumps. Consequently, our classification holds without any assumption of local constancy of the characteristic tuple.