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) classifying 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 under the appropriate pullback. The proof uses 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. For left-invariant metrics on step 2 Carnot groups, this tuple is the growth of the osculating flags of the Jacobi curves of generic normal extremals, i.e., the lengths of the columns of their Young diagrams starting from the second column. The compatible graphs are used where this tuple is locally constant; the resulting bracket relations extend by continuity, so our classification requires no local constancy of the tuple.